Patent Publication Number: US-10788634-B1

Title: Evolute tester for optical surfaces

Description:
FIELD OF THE INVENTION 
     The present invention is directed to testing and characterization of optical surfaces, including concave, flat, convex, and non-conic optical or reflective surfaces. It does not require that a master surface be first produced. 
     BACKGROUND OF THE INVENTION 
     The Frenchman Leon Foucault in 1858 invented a method of testing concave optical surfaces using a pinhole, light source, and knife-edge. Refinements of this method permit characterization of such surfaces to a tolerance of about 1/10 wavelength of sodium light (the equivalent of two millionths of an inch) for optics having a focal ratio slower than about f/2.0, and have permitted amateur telescope makers to construct excellent telescopes using very minimal tooling. The focal ratio, or f/number, is the focal length divided by the diameter of the surface. “Fast” means a low f/number (big fast lens on a camera might be an f/1.5, small or slow lens, f/8 or so). The faster a lens, the quicker the exposure can be in photography: at f/1.5 you can shoot at 1/1000 of a second, but at f/8, the shutter needs to be open 29 times longer, which amounts to 1/35 of a second, to get the same amount of light on the film. “Fast” optics are much harder to get right. 
     The Italian Vasco Ronchi in 1923 invented a method based on the Foucault test, using a coarse grating instead of a knife-edge. It offered an alternative to, and an incremental improvement in accuracy over, the Foucault test. It, too, tests only concave optical surfaces. 
     In 1929 the Argentinian astronomers Gaviola and Platzeck invented the so-called Caustic test, which took measurements involving radii of curvature. This was publicized by Irvin Schroader in Albert Ingalls&#39; book, Amateur Telescope Making Book Three, published in 1953. Schroader claimed that the Caustic test could detect errors in a concave reflecting optic on the order of 1/100 wavelength of sodium light, though this is not true for optics faster than about f/3.3. It must be emphasized that this test also only works on concave surfaces. The term “caustic” here is a misnomer, first used by Platzeck and Gaviola, propagated by Schroader, and carried on by those who followed. The definition of the term is as follows: 
     Caustic (Optics): a surface to which rays reflected or refracted by another surface are tangents. Caustic curves and surfaces are called catacaustic when formed by reflection, and diacaustic when formed by refraction. 
     The actual test proposed by Platzeck and Gaviola was based on centers of curvature, not the envelope of reflected rays, and thus bears a loose relationship to the evolute of a curve. 
     Evolute: 
     The locus of centers of curvature of a curve. Equivalently, the evolute is the envelope of normals to the curve. 
     Interferometric tests have also been devised, but they require the preparation of a master surface against which comparison can be made, as does the testing of non-concave surfaces. Such master surfaces must be ground and polished to an extremely high degree of precision; the work is not economically feasible unless multiple surfaces of a kind are to be produced. The production surfaces are then tested via interference fringes with relation to the master. 
     There remains a need for a way to test and characterize convex, flat, and non-conic optical or other reflective surfaces without requiring production of a master surface. 
     SUMMARY OF THE INVENTION 
     The invention pertains to the testing and characterization of optical or other reflective surfaces. The invention provides a method of testing and characterization which works equally well on concave, flat, convex, and non-conic optical surfaces, and which does not require that a master surface be first produced. The method is automatic and requires little human intervention. It eliminates the need for fallible human judgment of the character and darkness of shadows. It provides an extremely high degree of accuracy, and provides repeatability of measurements within a minuscule tolerance of error. 
     The imaging properties of an optical device are primarily dependent on the character and characterization of its surfaces, and secondarily on the reflective and refractive properties of the elements of the device that contain the surfaces. Accuracy in the characterization or measurement of such surfaces determines the imaging properties, and has been the aim of every method of optical testing ever devised. 
     An automated method of ascertaining the figure of an optical surface by determining the evolute of the figure is disclosed. The method tests surfaces automatically, deterministically, and repeatably via orthogonal reflection by ascertaining the evolute of the surface&#39;s figure along multiple diameters of the surface. It is not limited to concave optical surfaces, but may be applied to convex and flat surfaces, and is not limited to second-degree (conic section) curves. 
     The implementation of this method desirably requires:
         An assembly, hereafter called the Laser Head, comprising:   A laser whose most salient property is a narrow beam width;   A beam splitter;   A detector for a reflected laser beam which has a very narrow angle of detection;   A frame for holding the laser, detector, and beam splitter in precise alignment;   A measuring device, hereafter called the X-Unit, for ascertaining the distance of the detector from the surface under test;   A device, hereafter called the Slope Unit, for varying the direction in which the laser head&#39;s laser beam projects with respect to the optical axis of the surface under test, also in small increments, with a means of measuring the slope of this direction (“m”) (This is in essence a precision goniometer);   A device, hereafter called the Intercept Unit, for changing the position of the laser head in a direction perpendicular to the aforementioned optical axis, and doing so in small increments, with a way to measure this position; this ascertains the Y-coordinate at which each measurement of the slope unit is taken;   A computer and a software program to vary and record the above quantities, to ascertain and record the intensity of the reflection of the laser beam from the surface under test, and to calculate the straight line along which the laser beam returns orthogonally from the surface to the detector through the beam splitter, given X, Y, and the slope, m;   Electronic components to interface the computer to the aforementioned devices, via a Universal Serial Bus or other means;   A test stand capable of rotating the optic about the aforementioned optical axis, either automatically under program control or manually, but in either case repeatably; and       

     A systematic method for marking the optic such that the rotational positions are recoverable and repeatable for comparison with subsequent tests. 
     The nature, features, and advantages of the present invention will become appreciated as the same become better understood through reference to the specification, claims, and appended drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1 and 1A  are overall views of a possible embodiment of the invention, showing three major sub-assemblies and a fourth, minor, one. A test stand with an optic mounted on it is also shown. The figure shows the full assembly of the Intercept Unit with the Slope Unit mounted on it, and the Laser Head mounted on the Slope Unit. The X-Unit is also shown in its simplest form, a dial gauge. 
         FIG. 1B  shows the test stand with optic mounted on it from a frontal view. 
         FIG. 2  illustrates an embodiment of the laser head assembly from a viewpoint to the left of the laser. 
         FIG. 2A  is an exploded view of the laser head assembly from a slightly different perspective, this time to the right of the laser. 
         FIGS. 3A, 3B, 3C, and 3D  show an embodiment of the Slope Unit using a stepper motor, leadscrew, and position scale with reader. Other embodiments are possible. 
         FIG. 3A  shows the carriage, stepper motor, rails, and swivels of the Slope Unit for ascertaining the slope “m” of the emitted and returning beam. The Slope Arm ( 46 ) is shown here in a slightly positive position. 
         FIG. 3B  shows the same Slope Unit with the Slope Arm ( 46 ) in a maximum negative position, from a viewpoint to the left of that of  FIG. 3A . 
         FIG. 3C  is an exploded view of the Slope Arm ( 46 ) and associated attachments thereto obliquely from its underside, with several parts which cannot be seen in the previous two figures. 
         FIG. 3D  is an exploded view of the Slope Arm ( 46 ) and attachments shown obliquely from above. 
         FIG. 4  shows an embodiment of the Intercept Unit as a standalone entity using a stepper motor, leadscrew, and position scale and reader, while  FIG. 4A  is an enlargement of the measurement unit thereof and  FIG. 4B  is a perspective view of an underside of a spacer block that forms a part of the Intercept Unit. Other embodiments are possible. 
         FIG. 4C  shows an exploded view of the three major sub-assemblies of the Intercept Unit, viewed unexploded in  FIG. 4 . 
         FIG. 5  is a depiction of a curve with its evolute; the evolute being expressed as an envelope of perpendiculars to the curve. 
         FIG. 6  is an enlargement of a small area of  FIG. 5 , showing three of the perpendiculars to the curve and their two relevant intersections. 
         FIG. 7  is a block diagram of the electronics that interfaces the steppers, sensor, and digital readouts to the Microprocessor and thence to the control computer. It is an overall picture of the mechanics and electronics of the invention. 
         FIG. 8  is a flowchart of the overall software algorithm. No distinction is made between Microprocessor and control computer functions. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The present application provides systems and methods for testing and characterization of optical or other reflective surfaces. The benefits are most likely seen in the optical field, which is a multi-billion dollar per year industry. However, the same techniques may be useful for other reflective surfaces such as a radio antenna, an automotive fender, a curved window, a sculpture, etc. The techniques disclosed herein will thus work on any shiny surface, and the term reflective surface will thus be understood to encompass optical curved surfaces as well as other curved surfaces. 
       FIGS. 1 and 1A  show an overall view of the invention. The X-unit  18  is simply a dial gauge, the extreme extension of which is a known offset from the emitter of the laser  26  ( FIG. 2 ) in the laser head  20 . In the position shown, it is used to ascertain the distance to a point on the optical surface under test (i.e., a lens or mirror surface), which is shown as item  21 . The laser head  20  is mounted on the slope arm  46  of the slope unit  22  ( FIG. 3A ). The slope unit  22 , in turn, is mounted on the intercept carriage  104  ( FIG. 4A ) of the intercept unit  24  via the four clearance holes in the slope unit base  52  (also seen in  FIG. 3A ). The optical surface  21  is mounted rigidly but removably on the test stand  300 . The test stand  300  is so designed that the optical surface  21  may be marked so that it can be removed for work and replaced in the same position. 
     In  FIG. 1B  the optical surface  21  mounted on the test stand  300  is shown in a frontal view. The test stand  300  permits the optical surface  21  to be rotated about its center C/L and locked repeatably in any of several angular orientations with a pin or other such locking mechanism, since the evolute obtained during a test is for a single line from the center of its surface to its outside edge. At least two of these evolutes are required to characterize the surface, since it may not be a surface of revolution, but may possess astigmatism. This allows characterization of astigmatism in the surface, whether desired by design or needing to be eliminated as undesirable. Thus we may test surfaces that are surfaces of revolution and surfaces that are not. A dial gauge (not shown) may be mounted to the vertical arm of the test stand  300  so that the center of the optical surface  21  may be made coincident with the center of rotation of the stand, which is set at the same height as the center of the laser head  20 . Details of the stand are given here only for clarity, and one of skill in the art will understand that other configurations of mounting stands may be used 
     In  FIG. 2  is shown greater details of one version of the laser head  20  seen in  FIG. 1  which projects a laser beam  42  to the optical surface  21  under test. In this embodiment, a laser pointer  26 , modified to accept external power through a miniature phone plug  38 , is mounted in a holed ball  34  with its emitter coincident with the center of the holed ball  34 . The holed ball  34  is socketed in a spherical recess of the dual socket  32 , and held in place within a spherical recess of socket cap  36 . The beam  42  of the laser is projected to the right through beam splitter  30 , to the optical surface  21  under test ( FIG. 1 ). The beam returned from the surface is reflected by the beam splitter&#39;s internal surface  31  to the sensor  28 , which outputs a signal on the sensor pins  40  indicating the strength of the returned beam. 
     In  FIG. 2A  an exploded view of the laser head is provided, from a viewpoint to the right of that of  FIG. 2 . This reveals the second holed ball  34 ′ and the pinhole  43  which are not visible in  FIG. 2 . The sensor  28  is mounted in the second holed ball  34 ′, which is socketed in the other spherical recess of the dual socket  32 , and held in place by the second socket cap  36 ′. 
     In  FIGS. 3A, 3B, 3C and 3D , an embodiment of the slope unit  22  is shown. The slope unit  22  serves as a high-precision goniometer, an instrument for the precise measurement of angles, capable of measuring the slope of a line to approximately one part in 200,000 or a little less than one arcsecond. This embodiment uses a precision stepper-motor  68  ( FIGS. 3A and 3B ) for driving it, and a magnetic scale  90  ( FIGS. 3B, 3C, and 3D ) and scale-reader head  94  ( FIGS. 3C and 3D ) to ascertain position information. Other embodiments could use different driving methods, e.g., a voice coil for higher speed, and/or different measuring methods, e.g. a laser interferometer for higher precision. 
       FIG. 3A  shows the slope unit  22  from above with the slope arm  46  with attached slope rail  56  pointing in a slightly positive slope (CCW) about a slope axis  47 . The position of slope arm  46  is read by read head  94  on scale  90 , which is seen in an exploded view from its underside in  FIG. 3C .  FIG. 3B  shows the same unit with the slope arm  46  in its approximately maximum negative position (CW). The slope arm  46  pivots in both directions about the axis  47  ( FIG. 3A ) which is vertical to the plane of the page.  FIG. 3C  shows an exploded view of the slope arm  46  with the connecting hardware which cannot be seen in  FIGS. 3A and 3B .  FIG. 3D  shows attachment points which are otherwise invisible in the other three figures. 
     The slope arm  46  is rigidly attached to the slope rail  56  ( FIG. 3C ). Via adapter  76  and ball bearing  78  ( FIGS. 3C and 3D ) it pivots concentrically about the axis  47  ( FIG. 3A ) in the pocket of arm bearing block  48  ( FIGS. 3C and 3D ). Referring to  FIGS. 3A and 3B , the slope arm  46  and slope rail  56  are driven, as shown by movement arrows  59 , by the slope drive unit which comprises leadscrew  44 , slope drive base  66 , stepper motor  68 , drive carriage  58 , and cable connector  70 . The leadscrew  44  drives drive carriage  58 , which in turn drives the positive (CCW) and negative (CW) swing of the slope arm  46  and slope rail  56  (shown by movement arrows  53 ), by means of ball bearing  72  ( FIGS. 3C and 3D ). Ball bearing  72  is mounted to slope arm drive element  60  ( FIGS. 3C and 3D ), which is attached to slope rail drive carriage  74  ( FIG. 3C ) and socketed in the pocket in drive carriage  58 &#39;s upper surface ( FIG. 3D ). Slope rail drive carriage  74  rides in slope rail  56 . 
     The slope arm driven element  62  is attached to rail carriage  80  ( FIG. 3C ) which also rides in slope rail  56 . Ball bearing  82  ( FIG. 3C ) is mounted on the post of this driven element  62  and drives measurement element  64  via the pocket in its upper face ( FIG. 3D ). Measurement element  64 , in turn, is attached to rail carriage  84 . On the same side of measurement element  64  is attached connector block  86  ( FIG. 3D ), which is in turn attached with a position-adjustable mounting method to reader head mounting bracket  92 . This assembly carries and holds in place position encoder reader head  94 , which rides on, and reads position data from, position encoder scale  90  ( FIGS. 3C and 3D ). 
     Position encoder scale  90 , encoder scale mounting bar  88 , and measurement rail  54 , along with slope unit base  52 , form a unit, called the Slope Measurement Base Unit, with position encoder scale  90  and scale mounting bar  88  fastened underneath slope unit base  52  and measurement rail  54  fastened on top of it. Rail carriage  84  ( FIGS. 3C and 3D ) rides on measurement rail  54  and is attached to the underside of measurement element  64 , as described just above. Thus rail carriage  80 , slope arm driven element  62 , ball bearing  82 , measurement element  64 , connector block  86 , rail carriage  84 , reader head mounting bracket  92 , and reader head  94  form a unit which moves on both measurement rail  54  and position encoder scale  90  simultaneously at the impetus of measurement arm  46  and slope rail  56 . Position is read from position scale  90  by reader head  94  and transmitted to the electronics of the device via position encoder cable  96  ( FIG. 3C ), shown as a stub here. 
     The extension of slope arm  46  and slope rail  56  from slope arm drive element  60  to slope arm driven element  62  multiplies precision by a factor that is determined by the ratio of the distance between the axis  47  and the center of the pocket in measurement element  64 , to the distance between the axis  47  and the center of the pocket in carriage  58 . Distances read via position encoder reader head  94  are directly convertible to the slope of the slope arm  46  and attached slope rail  56 . The arm bearing block  48  is rigidly attached to the arm bearing block extension  50  which is rigidly attached to the base  66  of the slope drive unit. The base  66  of the slope drive unit is rigidly attached to the Slope Measurement Base Unit. 
     Motions of the various components of the slope unit are shown by movement arrows  53 ,  59 ,  61 ,  63 , and  65  ( FIGS. 3A and 3B ). Arc  53  indicates the swing of slope arm  46  with slope rail  56  and all components attached, around axis  47 . Impetus for this swing is given by carriage  58 , whose motion is shown by movement arrows  59 . This also causes motion of drive element  60  in the directions shown by movement arrows  61 . Additionally, the motions of rail carriage  80  are shown by movement arrows  63 , and the simultaneous motions of rail carriage  84  are shown by movement arrows  65 . 
       FIGS. 4 and 4C  show an embodiment of the intercept unit  24  in assembled and exploded views, respectively. In this embodiment, as with the slope unit  22 , a drive unit is provided by a stepper motor, and position measurement is done with a magnetic scale and reader. Again, as with the slope unit, other embodiments could use other units and methods of driving and measurement, such as voice-coil and laser interferometer, for more speed and more precision. In  FIG. 4A  is shown a detail of the unit comprising items  114 ,  122 , and  128  (twice) that rides on rail  124 . Detail circle “ 4 A” in  FIG. 4  shows Measurement Unit  102  and associated measurement hardware. 
     Detail “ 4 B” in  FIG. 4  shows the spacer block  114  and spacer base  122  with two rail carriages  128 , which reduce backlash from, and provide compliance with, the motion of the intercept carriage  104 . These rail carriage units, manufactured by the Igus Corporation of Cologne, Germany, are identical to rail carriage units  74 ,  80 , and  84  (in  FIG. 3C ), and ride, as shown in  FIG. 4C , on extension rail  124 . 
     The intercept unit  24  comprises three major sub-assemblies. These are the extruded base  100  to which is mounted the second sub-assembly, the overall ballscrew unit  98 , and the third sub-assembly, the measurement unit  102 . These are shown in  FIG. 4A  and in exploded view in  FIG. 4C . 
     The extruded base  100  consists of several lengths of aluminum extrusion fastened together as shown in  FIG. 4C , whose purpose is to act as a framework for the other two sub-assemblies, overall ballscrew unit  98  and measurement unit  102 , and to hold extension rail  124 , as described above. 
     The Overall Ballscrew unit  98  in this embodiment comprises a  32 ″ ball screw assembly with stepper-motor drive. It is mounted on lower base  118  and upper base  116 , which are used to fasten it to the extruded base  100 . The stepper-motor drives the screw which causes the intercept carriage  104  to move in the directions of movement arrow  105 . 
     The Measurement Unit  102  comprises the intercept encoder scale  106 , the intercept reader head  108 , the intercept reader head bracket  110 , and the intercept reader head cable  112  (shown as a stub). A detail of the Measurement Unit  102 , with its components, is shown in  FIG. 4A . 
       FIG. 5  shows the method of constructing the evolute of a conic-section curve  130  (in this case an ellipse) from orthogonals  132  to the curve. Each of these orthogonals  132  is projected to an intersection with the next one above it. The first orthogonal is ascertained by a method that involves minimal operator set-up and intervention. Each orthogonal after the first one is found by first moving the intercept-axis to a new position: the stepper-motor of the intercept unit  24  is stepped, under the control of the Microprocessor, while the output of the sensor  28  on the sensor line  163  is monitored. The sensor  28  provides an analog signal proportional to the intensity of the reflection  174 , which is sent via sensor line  163  to an analog-to-digital converter, part of the Microprocessor  178  (see electronics block diagram of  FIG. 7 ). The Microprocessor monitors the converted digital signal until a predetermined level of fall-off of intensity is found (this signals that an unobserved portion of the surface being tested  176  has been brought under observation); then the slope unit  22 , while again monitoring the sensor line  163 , is stepped through the sensor&#39;s maximum value, to find the value of slope that returns the brightest reflection of the laser beam at that intercept unit  24  position. The brightest reflection, measured as described via sensor line  163 , always occurs when the beam is orthogonal to the surface, i.e., when the reflected beam  174  is tangent to the evolute. The intercept-axis value at this point is ‘b’ and the slope-value is ‘m’, and the equation for the orthogonal is y=mx+b. Since the position of the slope unit  22  at which the sensor&#39;s maximum value occurs must be passed to insure that it is a maximum, the final step in this process is to return the slope unit&#39;s position to that position of maximum sensor return, at which the process can begin again. The envelope  136  of the fifteen orthogonals shown approximates the evolute of the curve  130  to a degree of precision determined by how closely they are spaced. The dashed lines  134  on the left side of ellipse  130  are tangents to the ellipse at the points of origin of the orthogonals. It should be noted here that the evolute  136  of the curve  130  is as shown, regardless of whether the surface reflects light to the right as a concave surface or to the left as a convex surface. After each orthogonal is found, the slope-value of the slope-unit  22  is returned to the place where the intensity maximum was recorded, so that the process can be repeated. The process is terminated when the laser beam passes the outer edge of the surface under test. 
       FIG. 6  shows three of the orthogonal lines of  FIG. 5  and indicates how the interpolation of their two intersections is done to give a point on the evolute. Uppermost orthogonal  144  intersects middle orthogonal  142  in point  146 . Middle orthogonal  142  intersects lower orthogonal  140  in point  148 . The center  150  of the segment between points  146  and  148  is taken as the center of curvature of the ellipse  130  at the point where the orthogonal  142  intersects it. In practice, the distance between points  146  and  148  will be on the order of a small fraction of a millimeter. Any point  150  ascertained by the interpolation method described above may be regarded as the center of curvature of the surface being characterized at the point from which the laser beam is reflected, as it will fall within a microscopic distance from the said center. It is therefore, within a very small tolerance, a point on the evolute of the surface. There are naturally some inexactitudes in (1) the point where the laser beam strikes the surface (because the beam makes a spot of finite width) and (2) the exact position of the point ascertained by the interpolation (due to the fact that there is a finite length between the two intersections). These two inexactitudes are minimized by minimizing the spacing of the points on the optical surface by the incremental “brightest reflection” method described herein, for any given beam width, but will vary for different optical surfaces. 
       FIG. 7  is a block diagram of the electronics that interfaces the steppers, sensor, and digital readouts to the computer. It is an overall picture of the mechanics and electronics of the invention. 
     The Control computer  152  sends commands to the microprocessor  178  (a micro-controller, for example, available from Arduino of Somerville, Mass.) and receives data from it. The microprocessor  178  commands the two Stepper motor controllers  154  and  164  (e.g., S-6 controllers available from Compumotor, a division of Parker Hannifin Corporation of Charlotte, N.C.), and receives positioning information from the two DRO Scale readers  162  and  172 , used in this embodiment of the invention (available from DRO Pros, 4992 Alison Parkway, Vacaville, Calif.), and intensity information on the reflected laser beam  174  from Sensor  28  via Sensor Line  163 . Stepper motor controllers  154  and  164 , with, respectively, stepper motors  156  and  166 , in turn drive the two ball screws  158  and  168  that control, respectively, the slope of the laser beam  42  and the y-intercept of its origin. The ball screw  158  of the Slope Unit  22  is rigidly connected to the DRO scale  160 , so that the position of the carriage of the ball screw  158  is read by the DRO scale reader  162 , and transmitted back to the Microprocessor  178 , from which it is relayed, with y-intercept information, to the control computer  152 , and where it is interpreted as the slope of the beam. The laser head  20 , mounted rigidly to the slope arm  46  ( FIG. 3A ) casts a narrow beam  42  ( FIG. 2 ) onto the surface being tested  176 , which is reflected  174  from that surface back to the sensor  28  of the laser head  20 . The sensor  28  senses a maximum return when the laser beam  42  is oriented orthogonally (perpendicularly) to the optical surface being tested  176  (also seen at  21  in  FIG. 1 ). 
     The Microprocessor  178  drives the ball screw  158  of the slope unit via Stepper motor controller  154  and stepper-motor  156  until the maximum return from the sensor  28  is sensed via the sensor line  163 . The slope value “m” at this point, together with the intercept-axis position “b”, fully define the line orthogonal to the optic at that intercept-axis position, as described in paragraph 0056 above. 
     The control computer  152  then drives the intercept unit&#39;s ball screw  168  via Stepper motor controller  164  and stepper-motor  166  until the return on the sensor line  163  falls off significantly from the maximum already sensed. 
     The cycle of the previous two paragraphs then repeats, until the laser beam passes the edge of the surface under test. 
     As noted above, the physical test stand  300  for the optical surface  21  is configured so that the optical surface  21  may be tested repeatedly in multiple angular orientations about its center, removed for corrective work, and replaced in an identical repeatable position and orientation on the test stand  300  for further testing. 
       FIG. 8  is a flowchart depiction, in broad general strokes, of the software algorithms required to drive the invention. Input parameters, setup of the optic to be tested and characterized, and evaluation of the results, are all operator functions. Communication between the Microprocessor and the control computer is assumed rather than explicitly diagrammed, and no dichotomy of their separate functions is indicated as these will be obvious to a practitioner skilled in the field. 
     The operator first starts the Microprocessor software on the Microprocessor, then starts the control computer software on the control computer. At the control computer the operator then ( 200 ) enters the requisite parameters on the optic under test (e.g., whether it is convex, concave, or non-conic, its nominal radius of curvature, its diameter, and such other parameters as may be necessary). The operator then commands initialization of the Microprocessor and the invention via the control computer. 
     When this is complete, the Microprocessor informs the control computer, which then, under program control ( 202 ,  204 ) reminds the operator to place, center, and align, the optic to be tested on a test stand. When the operator signals that this is done ( 204 ), the control computer commands the Microprocessor to back off the intercept axis so that the cusp of the evolute may be found, and next, to find the first orthogonal ( 206 ). When the Microprocessor has done so, it transmits the slope and intercept data back to the control computer. 
     The control computer receives and stores these data, and then, in a software loop, commands the Microprocessor to find the next orthogonal ( 208 ). Each time it attempts this, the Microprocessor arrives at one of four scenarios; they are (1) it may find and transmit another pair of orthogonal data to the control computer ( 240 ,  242 ); (2) it may finish the test normally ( 210 ) (i.e., it may reach the extent of the optic under test); (3) it may find that the optic is a flat ( 220 ,  222 ); or (4) it may encounter an error which prevents the test from continuing ( 230 ). (1) and (2) are the normal results of the method; (3) will happen on the second orthogonal; and (4) is an abnormal termination which may occur for a variety of causes. 
     Previous methods used a mask over the optical surface with holes spaced evenly across the diameter of the surface or the measuring stick of A. W. Everest. (Everest promulgated the Everest scale in Amateur Telescope Making Advanced, edited by Albert G. Ingalls, on page 21, Publications of the Astronomical Society of the Pacific (1937). It works better than the mask, and makes observing the optic easier.) In the current method, the system moves from one intercept-axis position that gives the brightest reflection to a new one where the reflection has fallen off to close to nothing, and then swings the slope-axis until the reflection from the new position is maximum; then repeats the process. This may give a variable spacing of points across the optical surface, but guarantees that for the given beam width they can be no closer to each other. This is done as part of step ( 208 ) “find the next orthogonal” code, in the flowchart of  FIG. 8 . Said spacing will also vary depending on the character of the surface being tested. 
     In case (1) (it found the next orthogonal) the Microprocessor transmits slope and intercept data to the control computer which records them and ascertains where this orthogonal intersects the previous one, as diagrammed in  FIG. 5  and  FIG. 6 . If there have been two or more orthogonals found before this one, the intersection of the two previous orthogonals, along with the intersection of the current one with the immediately previous one, is used to interpolate a point of the evolute, as follows: if {(b 1 ,m 1 ), (b 2 ,m 2 ), (b 3 ,m 3 )} are the first, second, and third orthogonals, respectively, then the point P(x p , y p ) on the evolute is given by 
     
       
         
           
             
               
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                     ( 
                     
                       
                         b 
                         2 
                       
                       - 
                       
                         b 
                         3 
                       
                     
                     ) 
                   
                 
                 
                   2 
                   ⁢ 
                   
                     ( 
                     
                       
                         m 
                         3 
                       
                       - 
                       
                         m 
                         2 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   
                     
                       b 
                       2 
                     
                     + 
                     
                       b 
                       3 
                     
                   
                   2 
                 
                 . 
               
             
           
         
       
     
     When scenario (2) happens, the test is finished and the (x p , y p ) points may be plotted and compared with the calculated evolute. 
     We calculate the desired evolute as follows: assume that the curve whose evolute we wish to calculate is g(y), some function of y, with first and second derivatives g′(y) and g″(y) respectively. We may express the evolute as a parametric equation: 
     
       
         
           
             
               
                 C 
                 
                   y 
                   ⁡ 
                   
                     ( 
                     
                       x 
                       , 
                       y 
                     
                     ) 
                   
                 
               
               = 
               
                 
                   1 
                   
                     
                       g 
                       ″ 
                     
                     ⁡ 
                     
                       ( 
                       y 
                       ) 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           
                             g 
                             ′ 
                           
                           ⁡ 
                           
                             ( 
                             y 
                             ) 
                           
                         
                         2 
                       
                       + 
                       1 
                       + 
                       
                         
                           g 
                           ⁡ 
                           
                             ( 
                             y 
                             ) 
                           
                         
                         · 
                         
                           
                             g 
                             ″ 
                           
                           ⁡ 
                           
                             ( 
                             y 
                             ) 
                           
                         
                       
                     
                     , 
                     
                       
                         y 
                         · 
                         
                           
                             g 
                             ″ 
                           
                           ⁡ 
                           
                             ( 
                             y 
                             ) 
                           
                         
                       
                       - 
                       
                         
                           
                             g 
                             ′ 
                           
                           ⁡ 
                           
                             ( 
                             y 
                             ) 
                           
                         
                         · 
                         
                           ( 
                           
                             
                               
                                 
                                   g 
                                   ′ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   y 
                                   ) 
                                 
                               
                               2 
                             
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
           
         
       
     
     so that for any point (g(y),y) on the curve g(y), the corresponding center of curvature, or equivalently, the point on the evolute P E (x E , y E ), is 
               x   E     =       g   ⁡     (   y   )       +           (       g   ′     ⁡     (   y   )       )     2     +   1         g   ″     ⁡     (   y   )                         y   E     =     y   -         g   ′     ⁡     (   y   )       ⁢           (       g   ′     ⁡     (   y   )       )     2     +   1         g   ″     ⁡     (   y   )                   
which may be plotted on the screen of the control computer and compared both visually and numerically with the evolute measured as above.
 
     The desired curve of the surface is determined by the optical properties for which it is designed, and is therefore known by design. We calculate the evolute from the desired curve of the surface, by the mathematical formulae shown above. We make the surface, by grinding and polishing it, with standard optical methods. We mount the surface on the test stand and measure the evolute of the surface with the tester by the methods described herein, and get a very close approximation to its actual, physical evolute; then we compare the measured and calculated evolutes, ascertain where they differ by more than the allowed tolerance, and, if it does not fall within the allowance, do further optical work (i.e., grinding and polishing) on the surface to bring it within tolerance. We repeat the cycle of measurement and work until the surface is within tolerance. 
     CONCLUSION 
     Throughout this description, the embodiments and examples shown should be considered as exemplars, rather than limitations on the apparatus and procedures disclosed or claimed. Although many of the examples presented herein involve specific combinations of method acts or system elements, it should be understood that those acts and those elements may be combined in other ways to accomplish the same objectives. With regard to flowcharts, if present, additional and fewer steps may be taken, and the steps as shown may be combined or further refined to achieve the methods described herein. Acts, elements and features discussed only in connection with one embodiment are not intended to be excluded from a similar role in other embodiments. 
     As used herein, “plurality” means two or more. As used herein, a “set” of items may include one or more of such items. As used herein, whether in the written description or the claims, the terms “comprising”, “including”, “carrying”, “having”, “containing”, “involving”, and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of” and “consisting essentially of”, respectively, are closed or semi-closed transitional phrases with respect to claims. Use of ordinal terms such as “first”, “second”, “third”, etc., in the claims to modify a claim element does not by itself connote any priority, precedence, or order of one claim element over another or the temporal order in which acts of a method are performed, but are used merely as labels to distinguish one claim element having a certain name from another element having a same name (but for use of the ordinal term) to distinguish the claim elements. As used herein, “and/or” means that the listed items are alternatives, but the alternatives also include any combination of the listed items. 
     Those skilled in the art will appreciate that various changes and modifications may be made to the preferred embodiments, the invention in its broader aspects is not limited to the specific details, representative devices, and illustrative examples shown and described.