Author: csadmin

Using The Point Source Microscope (PSM) To Find Conjugates Of Parabolic And Elliptical Off-axis Mirrors

Written by csadmin on April 12, 2016. Posted in PSM Applications, Published Papers. No Comments on Using The Point Source Microscope (PSM) To Find Conjugates Of Parabolic And Elliptical Off-axis Mirrors

ABSTRACT

Although the PSM is primarily an alignment instrument, it can also be used to determine the conjugates of parabolic and elliptical off-axis mirrors. By positioning the PSM at the sagittal and tangential foci of the mirrors, the conjugate distances of the mirror can be found using a laser range finder, for example. Knowing the sagittal and tangential radii of curvature (Rs and Rt), the vertex radius (Rv) is easily calculated. This information is used to verify that the mirror has been correctly manufactured and to aid in positioning the mirror in an optical system. Examples are shown of these steps.

1. INTRODUCTION

The alignment of off-axis aspheres is a subject that few people want to address because it is so different than aligning symmetric optics. An off-axis asphere has an optical axis, as opposed to a sphere that has no optical axis, just a center of curvature. Furthermore, the optical axis of the off-axis asphere usually does not pass through the surface of the mirror and certainly is not parallel to the normal to the surface at the center of the off-axis segment. On the other hand, off-axis aspheres are extraordinarily useful for many optical designs, and familiarity with them is essential for many optical systems.

This paper is an attempt to take some of the mystery out of off-axis parabolas and ellipses in terms of their geometry, and how to use their geometry to aid in their alignment. We will start by discussing some of the geometry that makes off-axis mirrors unique, and then describe how to find and measure the conjugates that are analogues to the center of curvature for a sphere. As opposed to a sphere with a center of curvature, an off axis asphere has two conjugates that define its axis, the two astigmatic foci at the centers of curvature of the tangential and sagittal radii of curvature. With the knowledge of how to find these conjugates, and a method of measuring the radii of curvature, it is possible to see that the mirror was manufactured correctly, and to see how the mirror should be aligned into the optical system for which it is intended.

2. GEOMETRY OF AN OFF-AXIS PARABOLA

Fig. 1 is a scale drawing of an off-axis parabola looking at a meridional section and along the optical axis (black vertical line). The off-axis section is the circle at the right edge of the pie section looking along the optical axis. It is aligned with the three vertical lines that defined the inner and out meridional extents of the off-axis section. The parent parabola is green and the blue circle has a radius equal to the vertex radius of the parabola while the purple circle segment, whose tangential radius is labeled Rt, is the best fit circle to the center of the off-axis segment. The light blue rays stopping at the optical axis define the sagittal radius of curvature, Rs.

The three radii are related by Rt = Rs^3/Rv^2 which makes it easy to find an unknown radius if the other two are known1. If a point source of light is placed on the axis of the parabola, a line image will form at the axis (light blue line perpendicular to the Rs ray). In general, the image will not be a line until the aperture of the off-axis section is masked, or stopped, down to a diameter where the coma in the off-axis asphere does not affect the astigmatism. How big a hole in the mask can be determined empirically, or an estimate of the coma may be made referring to Ref. 2.When the point source of light and the return line image are superimposed, the point source is on the normal to the surface at the center of the hole in the mask. The distance from the surface to the point source/line image is the sagittal radius, Rt. The astigmatic line image will lie in the meridional plane and is helpful in determining the azimuth of the vertex of the offaxis segment as well as determining the sagittal radius.

If the point source of light is moved back along the normal until the astigmatic line image is in focus it will be seen to be perpendicular to the meridional plane. The distance from the point source/line image here is the tangential radius, Rt. Since Rs = sqrt(Rv^2 + h^2) where h is the off-axis distance it is easy to find h.

3. GEOMETRY OF AN OFF-AXIS ELLIPSE

The geometry of an off-axis ellipse is a little more complex but the same basic geometry applies. Fig. 2 is a scale drawing of one of the parts we actually measured using the PSM and FARO arm. The green rays show the extent of the off-axis section relative to the foci to give a feel for the scale. The rays that bisect the rays going to the foci are those seen from the center of curvature of the off-axis segment. The right side of Fig. 2 shows the caustic region to give a better idea of the locations of the sagittal and tangential centers of curvature relative to the center of curvature of the vertex sphere.

As with the parabola, the sagittal foci lie on the axis of symmetry, or optical axis, while the tangential foci lie of the far side of the axis. From the way an ellipse focuses light from one focus to the other, we know the normal to the surface is the perpendicular bisector of the rays between the foci as shown. Just as with the parabola, we can find Rs and Rt, and using these values we can find Rv = (Rs^3/Rt)^1/2. For the ellipse we still need to find the conic constant, k, and the distance off-axis, h. To do this we make a mask as in the case of the parabola and measure Rs and Rt at 2 or 3 locations, h and ±delta h, where delta h is known. Two measurements will give an exact solution, while 3 or more will give a least squares solution using the relation that Rv^2 –k*h^2-Rs^2 = 01. The simplest method is to use 2 measurements at either edge of the off-axis ellipse and find the solution using the quadratic equation. One thing to remember is that delta h measured off the optic will be larger than it actually is, the distance from the axis of symmetry. Entering delta h as 1.4x less than the width of the optic is probably a good guess for a start.

Clearly with a concave ellipse this approach will get you close because the foci have to lie on the same line, the optical axis, as the sagittal foci. Also, for any given h, the line joining Rs and Rt lies on the normal to the surface. This will help locate the foci approximately. With this information it is then more precise to try to find the foci directly using the information derived from Rs and Rt. Using a point source such as the end of a single mode fiber at one focus and a microscope at the other focus it should be possible to locate the 2 foci to the precision necessary to meet system tolerances.

4. MEASURING THE SAGITTAL AND TANGENTIAL RADII

Using an autostigmatic microscope such as the PSM3, point the microscope at the mirror and locate the return image. Once the microscope is adjusted so the reflected point source is centered on the objective, focus the microscope to find either the sagittal or tangential focus. In general the reflected image will be too blurred to find a clean line image. Mask the mirror with an aperture small enough to produce a line return image. Adjust focus to get clean looking images such as that in Fig. 3.

When the microscope is well focused, bring a distance measuring device such as a laser distance measurer, or laser range finder4, up in front of the microscope and position it axially to get a focused Cat’s eye reflection off the rear of the distance measurer. Then take a distance reading to the mirror through the hole in the mask. Since most distance measurers calculate the distance from the rear surface of the instrument, the measurer reading is the sagittal or tangential radius, depending on which image is in focus. Fig. 4 shows how the measurement is made schematically.

Once these measurements have been made, the results may be compared with the prescriptions for the off-axis segments to see if the manufactured mirrors are to specification using the equations in the first part of the paper. It turned out in our case that one mirror was in spec and the other was not. Luckily it turned out that the specification was tighter than it needed to be and the mirror did not limit the performance of the system.

5. USE OF THE SAGITTAL AND TANGENTIAL FOCI FOR ALIGNMENT

There are at least two ways the focus information can be used for the alignment of off-axis mirrors. First, the measurements can be used directly by adding alignment features to the optical bench that the mirrors will be installed in. Since the two foci produce line images in reflection, we can use polished cylindrical artifacts to simulate the line foci. An inexpensive source of suitable artifacts are “plug gauges”5, cylinders about 50 mm long in diameters ranging from <1 mm to 25 mm or more. These are used the same way polished balls are as physical realizations of points to focus a PSM on, but in this case the cylinders are realizations of lines. Kinematic mounts should be made so 2 plug gauges are positioned where the sagittal and tangential foci are supposed to be located in the finished mirror system.

To align an off-axis segment, a PSM is focused on the line image formed at the axis of the plug gauge, the gauge is removed and the mirror adjusted so its focus appears in the same location as the line formed by the plug gauge. Fig. 5 shows a typical image on reflection from a plug gauge. The mirror can be located in 3 degrees of freedom, axially, one degree of translation, and one degree of rotation about the normal to the mirror surface. Once one focus is adjusted, the PSM is moved to the second plug gauge, and focused on it. The gauge is removed and the mirror adjusted by rotating around the first focus until second focus is correctly positioned laterally. This process may take some iteration because it may be difficult to make the mirror rotate around the first focus although with sufficient forethought the mirror mount could be designed with this adjustment built in.

A second method of alignment, an indirect one, is what we did. We located plug gauges at the two foci and then measured the locations of the gauges relative to fiducials on the mirrors. Fig. 6 shows how a plug gauge was adjusted to be at the same location as the line image formed by the mirror sagittal focus, in this case. This meant that when the mirrors were installed in their optical bench using the fiducials around the edges of the mirrors, we were effectively positioning the foci to their correct positions in space.

6. RESULTS OF MEASUREMENTS ON AN OFF-AXIS MIRRORS

The mirrors we measured were about 250 mm square and CNC machined out of 6061-T6 aluminum and then smoothed and polished by hand to a specular surface. One of the mirrors is seen behind the mask in the left hand part of Fig. 6. The ultimate use did not require the polish as they are to go in a terahertz wavelength (857 μm) radio telescope. On the other hand, we wanted to be sure the mirrors were sufficiently good to meet the telescope spec, and had been correctly machined and not deformed by the polishing.

Following the procedure described above we got the results shown in Tables 1 and 2. The standard deviation in terms of setting at the best focus of the line focus was about ±4 mm for the parabola and ±10 mm for the ellipse. The distance measuring laser gauge gave consistent readings within ±1 mm. Thus both mirrors had larger errors in their radii than could be attributed to measurement error. On the other hand, the radius errors amount to sag errors over the full aperture of 30 and 50 μm in the case of the ellipse, and 160 and 190 μm in the case of the parabola, compared with λ = 857 μm.

After reviewing these errors in manufacture in the optical design code it was felt that with a minor spacing adjustment the mirrors would work just fine as they were. This finding coupled with the knowledge of where the fiducials were relative to the radii of curvature meant that the system would perform as expected and could be successfully aligned.

7. CONCLUSION

We have shown how to determine the vertex radius of curvature and off-axis distance of off-axis parabolas by measuring the sagittal and tangential radii of curvature with an autostigmatic microscope such as the PSM. We have gone on to show how the conic constant may be determined as well by making multiple measurements of the radii of curvature for more general conics. This information is useful as part of incoming inspection to see if the optics ordered are the optics delivered. Further, we have shown how the locations of the radii of curvature can be used as a tool in aligning the offaxis mirrors into their optical systems using two different approaches to alignment.

REFERENCES

Smith, W., Modern Optical Engineering, 2nd Ed., McGraw-Hill, New York, 445-6 (1990)
Parks, R. E., Evans, C. J. and Shao, L., “Test of a slow off-axis parabola at its center of curvature”, Appl. Optics, 34, 7174-8 (1995).
See www.optiper.com
For example, see http://www.cpotools.com/bosch-dlr130k-digital-distance-measurerkit/ bshndlr130k,default,pd.html?start=1&cgid=bosch-locators-and-measurers
See, for example, http://www.mcmaster.com/#plug-gauges/=dbxtba

Method Of Alignment Using A Laser Tracker System

Written by csadmin on April 12, 2016. Posted in PSM Applications, Published Papers. No Comments on Method Of Alignment Using A Laser Tracker System

1. Introduction:

Laser trackers are an accurate and efficient tool for finding the locations of features in a threedimensional space but they rely on Spherically Mounted Retroreflectors (SMR) to return the laser beam to the tracker. If the feature cannot be contacted or it is not convenient to use an SMR another method must be used to follow the beam. We describe methods using a dual imaging and autostigmatic microscope for locating the features and two methods for tracking the microscope location depending on the type of tracker used. This converts a contact probe, large area CMM into a non-contact CMM by coupling a laser tracker with a dual purpose autostigmatic microscope. We begin with a brief description of the microscope followed by the alignment of the microscope to tracking and scanning laser metrology stations.

2. Point Source Microscope:

The imaging, autostigmatic microscope in question is called a Point Source Microscope (PSM)1 and has both a single mode fiber as a point source of light for the autostigmatic function and a LED behind a diffuser to provide Kohler illumination for imaging. The sources may be used independently or simultaneously. In the autostigmatic mode the PSM has lateral sensitivity of <1μm and similar axial sensitivity with an auxiliary lens for finding the centers of curvature of tooling balls or optics like mirrors and lenses. In the imaging mode the lateral sensitivity depends on the objective used but is in the neighborhood of a couple microns and the Cat’s eye reflection from a surface gives axial sensitivity to a similar resolution. It remains to show how to couple this non-contact ability to locate centers of curvature, the optically important feature of a lens or mirror for alignment purposes, to a high level of sensitivity to laser tracker systems that can determine locations in 3-dimensional space to similar sorts of accuracies.

3. Follower type laser tracking systems:

A follower type laser tracker is similar to a surveyor’s theodolite but is more sophisticated in that it is active. It sends out a laser beam along the telescope axis and the telescope will follow the beam if it is retroreflected back using a corner cube (generally a SMR that has the feature that the apex of the cube corner is accurately located at the center of the ball or spherical mount). The tracker will also measure distance using an interferometer mode if the SMR is moved from the tracker calibration SMR nest to a nest mounted on the measurand without breaking the laser beam. Fig. 1 shows how the follower tracker scheme works in general.

3.1 Alignment of the PSM to an SMR:

SMR’s are made in several convenient sizes, ½”, 7/8” and 1½” diameters, along with corresponding SMR “nests”, usually steel cones with a magnet at the apex to hold the SMR seated in the cone. A nest and corresponding SMR are mounted at a convenient location and the tracker is locked onto the SMR and its location in 3-D is noted. The SMR is removed from the nest and a grade 25 or better steel ball of the same diameter is placed in the nest so that its center is in the same location as the SMR. Then the PSM is adjusted so the focus of the objective is located at the ball center, something that can be done to <1μm in all three dimensions.

3.2 Alignment of the tracker to the PSM:

Attached to the PSM is a plate with a plane mirror in a mount that can be adjusted in tip, tilt and piston and a nest and an SMR. The plane mirror, shown schematically in Fig. 1, is nominally located half way between the objective focus and the SMR on the plate and perpendicular to a line between the focus and SMR. This arrangement is shown in Fig. 2. The mirror reflects the tracker laser beam back toward the SMR on the plate. The plane mirror is adjusted until the reflection from the SMR on the plate has the same apparent location as the SMR originally located at the PSM focus. The mirror is then adjusted such that the tracker sees the virtual image of the SMR at the objective focus. Now the assembly of PSM, plane mirror and SMR can be moved at a unit and the tracker will follow the PSM focus in all three dimensions as long as the PSM is not rotated so far that the tracker laser beam walks off the plane mirror.

4. Scanning type tracker system:

A scanning type tracker system uses several high speed rotating prism stations to sweep line images around a workspace. The prism scanners are accuracy synchronized with a clock. In the work space are detectors that resister when a beam sweeps by. Comparing the prism clocks and detector signals the detector locations can be accurately determined by time delay triangulation.

4.1 Alignment of the detectors to the PSM:

With the scanning system, three detectors are fastened to the side of the PSM facing two scanning prism stations. The PSM is focused on the center of a conveniently located steel ball. In this position the detectors are scanned by two scanning stations a known distance apart. Since three detectors on the PSM are scanned the plane of the detectors is determined relative to the scanning stations. The PSM with detectors attached is then moved to a different orientation while still being focused at the center of the ball and another set of data is taken to determine the plane of the detectors. This operation is repeated for a third orientation of the PSM while the PSM is centered on the ball. The three planes defined by the detectors will appear to rotate about the center of the ball, that is, a normal to each of the planes through the center of the ball are all equidistant as shown in Fig. 3.

This means the center of the ball, or focus of the objective, can be determined from the three measurements and once the distance from the center to the planes of the detectors is determined and recorded, any other position of the PSM can be related back to the center of the ball. This calibration is completely analogous to the initial use of a master ball and a touch probe on a Coordinate Measuring Machine (CMM) to establish the zero of the CMM coordinate system.

4.1.1 Using the calibrated PSM:

Once the center has been determined by this calibration, when the PSM is moved to any other location and the positions of the three detectors determined the location of the focus can be related to the center of the ball in all three coordinate directions. Any software that might be used with a CMM can also be used to analyze the data the scanner tracker gathers. This makes it possible to use the PSM precisely as a non-contacting probe with an apparent zero probe tip radius for any sort of coordinate measurements over large distances.

5.0 Conclusion:

We have shown how an autostigmatic microscope, and in particular, the Point Source Microscope (PSM) as a non-contact probe, can be used with laser tracker systems, either the follower type or the scanning type. In the first case an auxiliary mirror and tracker ball are used to make the tracker think it is looking at the focus of the PSM. In the second case, the PSM is calibrated in a way analogous to using the master ball on a CMM and then the PSM can be used as a zero radius probe tip for non-contact coordinate measurements over large distances. In both cases the PSM can be used either to find a center of curvature to 1 micron sensitivity in three dimensions or used in the imaging mode to locate a feature on a surface to 2 o3 microns in all three dimensions. The fact that the PSM images and can be set in all three coordinates to micron precision makes it a valuable part of an extension to contacting laser tracker systems.

6.0 References

1. Optical Perspectives Group, LLC, Tucson, AZ 85750, www.optiper.com.
2. J. Burge, et. al., Use of a commercial laser tracker for optical alignment, Proc. SPIE, 6676, 6676OE, (2007).

Swing Arm Optical Coordinate-Measuring Machine: High Precision Ground Aspheric Surfaces Using A Laser Triangulation Probe

Written by csadmin on March 10, 2016. Posted in Optical Testing, Published Papers. No Comments on Swing Arm Optical Coordinate-Measuring Machine: High Precision Ground Aspheric Surfaces Using A Laser Triangulation Probe

ABSTRACT

The swing arm optical coordinate-measuring machine (SOC), a profilometer with a distance measuring interferometric sensor for in situ measurement of the topography of aspheric surfaces, has shown a precision rivaling the full aperture interferometric test. To further increase optical manufacturing efficiency, we enhance the SOC with an optical laser triangulation sensor for measuring test surfaces in their ground state before polishing. The calibrated sensor has good linearity and is insensitive to the angular variations of the surfaces under testing. Sensor working parameters such as sensor tip location, projection beam angle, and measurement direction are calibrated and incorporated in the SOC data reduction software to relate the sensor readout with the test surface sag. Experimental results show that the SOC with the triangulation sensor can measure aspheric ground surfaces with an accuracy of 100 nm rms or better.

Subject terms: swing arm profilometer; profilometry; aspherics; optical testing; laser triangulation sensor; ground surface metrology.

Paper 120473 received Mar. 30, 2012; revised manuscript received May 16, 2012; accepted for publication May 30, 2012; published online Jul. 6, 2012.

1 Introduction

In the field of optical metrology for aspheric optics fabrication, most of the interferometric tests (visible spectrum range) that provide high-accuracy measurement are used after the test surface is polished. But at the polishing stage, the surface shape, or figure, correction is slow. It is desirable to measure the surface accurately during grinding to minimize figure errors and speed up fabrication.

Different techniques have been developed for testing ground surfaces. One traditional method is to use an infrared interferometer. However, as with visible interferometers, null optics are needed for measuring the aspheric shape. The design and alignment of the null optics are complex and time consuming. The limited infrared material choices and the nonvisible light make the use of the infrared interferometer even more difficult.

A laser tracker or laser tracker plus system¹ such as the one used for the Giant Magellan Telescope primary mirror segment is another way to measure a mirror during grinding. The basic idea uses a commercial laser tracker system with the spherically mounted retroreflector (SMR) touching the mirror surface. Since it is a point by point test, it takes time to collect a large number of samples.

There is a new prototype called SLOTS² based on reflection deflectometry using a long-wave infrared source scanning technique to measure the surface slope variation of ground surfaces. It is a simple, fast, low-cost, and non-null system that can measure surface slopes to microradian precision. The method is promising but needs further development.

The swing arm optical CMM (SOC),^3–5 developed at the University of Arizona, is a profilometer with a distance measuring interferometric sensor. It is used for in situ measurement of highly aspheric mirrors, and it has shown a performance rivaling full aperture interferometric tests. The interferometric sensor has high precision and a high data rate, but it only works for polished surfaces. A contact sensor, with a linear variable differential transformer (LVDT), has been used with the SOC to measure ground surfaces, but the measurement is time consuming, because the probe is picked up after each point to avoid scratching the surface.

Different types of sensors have been studied for ground surface metrology. There are some sensors based on the confocal principle⁶ with linearity at the submicron level. There are sensors based on laser triangulation that have a resolution of 10 nm and are insensitive to surface angular variation. The triangular sensors work on both polished and ground surfaces and were chosen and calibrated⁷ for use in our application. The laser triangulation technique has been used for many applications, such as inspection of free-form surfaces^8,9 and blind guidance.¹⁰ The measurement uncertainty due to speckle noise, the test lateral resolution, and the aspect of the test sensitivity to the surface texture distribution have been investigated.^11–13 For high-precision optical surface measurement applications like astronomical telescope mirror metrology, the only work reported are some simulations showing the potential measuring accuracy using different types of triangulation sensors.^14,15

This paper shows the results of calibrating a triangulation sensor and using it to measure large ground mirrors. The paper is organized as follows: In Sec. 2, we review the basic principle of the SOC and laser triangulation sensors. In Sec. 3, we describe the calibration of a triangulation sensor and show some metrology results compared to an interferometric null test of a highly aspheric mirror. Finally, conclusions are drawn in Sec. 4.

2 Principles

2.1 Principle of the SOC

The basic geometry of the swing-arm profilometer is shown in Fig. 1. A sensor is mounted at the end of an arm that swings across the optic under test such that the axis of rotation of the arm goes through the center of curvature of the optic. The arc defined by the sensor tip trajectory, for a constant sensor reading, lies on a spherical surface defined by this center of curvature. For measuring aspheric surfaces, the sensor that is aligned parallel to the normal to the optical surface at its vertex reads only the surface departure from spherical. The SOC uses this simple geometry with an optical, noncontact, interferometric sensor that measures continuously across the optic. The optic or test part is rotated in azimuth after each profile is measured. Figure 2 shows an example of the profiling pattern we generally use during a test. Since the arcs cross each other while the sensor scans the mirror edge to edge, we know the surface heights must be the same at these scan crossings. The crossing height information is used to stitch the profiles into a surface using a maximum likelihood reconstruction method.^3,16,17 Figure 3 shows the results of using the SOC and a full aperture interferometric null test to measure a 1.4-m diameter aspheric surface that has an aspheric departure of 300 μm. The SOC test shows excellent agreement with the interferometric test. The direct subtraction of the maps from the two methods, after alignment terms have been removed, shows a difference of only 9 nm rms, much of which appears to come from the interferometric test.

2.2 Principle of the Laser Triangulation Sensor

Laser triangulation is widely used in various applications to measure distances to objects. A common triangulation principle is to project a light spot on to the object and extract the distance information from the reflected or scattered light.¹⁴

High measuring rates, high spatial resolution, large measuring range, and zero applied force are significant advantages of the laser triangulation sensor over other types of distance measuring sensors.

There are two types of triangulation sensors normally used. The first one is an orthogonal sensor, in which the output plane is perpendicular to its optical axis. The second one has a tilted sensor, in which the output plane is tilted according to the Scheimpflug principle. Tilting of the output plane eliminates defocus and makes the sensor insensitive to the angular variations of the test surface due to the imaging effects.¹⁴ The sensor investigated and used on the SOC is the second type.

The geometry of the sensor measuring method is shown in Fig. 4. The incident beam with angle θ hits the surface at the sensor’s tip position O, where the sensor reads zero, and then reflects or scatters back and passes through an imaging lens inside the sensor. Finally, the beam falls on the detector of the sensor. The measuring axis is the bisector of the incident and reflected beam. By knowing the spot tip position and incident beam angle, we can trace the ray to O’ when the surface moves up to S2. The ray will reflect or scatter at the surface and be collected by the imaging lens if the distance d is within the sensor’s dynamic range. The sensor’s detector plane is tilted and is conjugate with the OO’ plane according to the Scheimpflug principle. The difference of the spot positions is a function of the displacement d, the projection beam angle, the magnification of the optical system, and the focal length of the imaging lens as described by Mikhlyaev.¹⁴ The commercial triangulation sensor is usually calibrated so that the output signal is linear with the test surface displacement.

3 Calibration

The sensor’s linearity, angular sensitivity, and scaling effect were measured experimentally. The zero position (tip position) of the sensor projected beam, the direction of the projected beam, and the measurement axis were calibrated and used as the input for the SOC data reduction. Finally, the system performance with the triangulation sensor was verified by measuring test surfaces with known shapes. The flow Fig. 1 Basic geometry of the swing-arm profilometer. chart in Fig. 5 shows the outline of the calibration steps.

3.1 Linearity

A distance measuring interferometer (DMI)¹⁸ was used to check the linearity of the triangulation sensor. The DMI has accuracy at nanometer level. The setup is shown in Fig. 6.

The DMI and the triangulation sensor were aligned to be able to measure simultaneously the motion from a double-sided flat mirror, which was mounted on a flexure. The flexure was driven with a voice coil and function generator to oscillate sinusoidally with a peak-valley motion of 2 mm at a frequency of a few Hz. One side of the double-sided flat mirror was specular as the target for the DMI, while the other side of the flat was ground for evaluating the triangulation sensor.

We budgeted a signal difference of 10 nm or less from the test system alignment and the motion effect. (The sensor has a 10 nm resolution.) The motion of the flexure with the flat mirror was designed and checked with an alignment telescope to show an angular variation of a few arc-seconds. The DMI and the triangulation sensor are aligned to each other to minimize the cosine errors and Abbe errors.

Figure 7 shows an example of the signals obtained from the DMI and the triangulation sensor.

The readout difference from the direct subtraction of the DMI and triangulation sensor data was ∼60 nm rms as shown in Fig. 8. The difference was dominated by the random noise from the environment and the ground mirror surface roughness, because the incident beam from the triangulation sensor moves across the mirror as it is oscillated. Later we show that when the sensor is mounted on SOC and tests a polished surface, a precision of 20 nm rms is achieved when an average of eight single measurements is used.

3.2 Calibration of the Geometry of the Sensor

3.2.1 Sensor tip location and the direction of the projected beam

When measuring a test surface, as seen in Fig. 9, the coordinates of the point of measurement are a function of the sensor position, the direction of the projected light beam relative to the surface, and the test surface shape. Following the SOC coordinate calibration concept described by Su et al.,4,5 we calibrated the relationship between the sensor nominal tip position when the sensor reads zero and the direction of the projected light beam relative to reference features on the sensor case, namely, three laser tracker spherically mounted retroreflectors (SMRs). When the sensor is installed on the SOC, the coordinate relationship is used to determine the sensor tip location and angle of the projected beam relative to the surface under testing. This is done by measuring the three SMRs’ locations with a laser tracker. Then at each scan position, the coordinates of the point of measurement on the test surface can be calculated with a simple ray tracing algorithm in the SOC data reduction code.

To perform the calibration, we used a point source microscope (PSM)¹⁹ and a laser tracker.²⁰ As shown in Fig. 10, a piece of 25-μm thick translucent shim stock was located between the sensor and the PSM. The shim stock was positioned normal to the sensor’s measuring axis. The sensor was mounted on an x, y, z stage that brought the sensor up to the shim stock until the sensor read zero. Then a PSM was brought up to focus on the backside of the shim stock, so the PSM was focused on the effective sensor measuring spot as seen through the shim stock. When the sensor and the PSM were in these positions, the locations of the three SMRs on the sensor were measured with the laser tracker. Then the sensor was moved out of the way, and the center of a solid steel ball was positioned at the focus of the PSM objective. Then the solid ball was removed from the SMR nest and replaced with an SMR.

The SMR was measured with the laser tracker as shown in Fig. 11. Now the laser tracker has information about the measuring spot location at the zero height reading of the sensor relative to the SMR nests on the sensor. By moving the shim stock to different sensor height locations of the projected beam and repeating the above procedures, several locations along the projected beam were measured, and the direction of the projection beam was determined. The thickness of the shim stock was backed out during the data processing. The accuracy of the calibration in terms of the SOC coordinates was determined to a few microns.

3.2.2 Sensor measurement axis

The readout of the sensor is the displacement d along the axis of the sensor as shown in Fig. 9. Thus it is important to calibrate the sensor measurement direction relative to the sensor axis and align the sensor axis to the SOC test system. This calibration was initially done using the setup in Fig. 6. After the DMI and the triangulation sensor were aligned to minimize the readout difference between the two sensors, we used a laser tracker to measure the three SMRs’ locations on the sensor. Then the double-sided flat mirror and the sensor were removed, and a reference flat mirror was put far from the DMI but normal to the DMI beam as shown in Fig. 12. We used a tracker ball touching the mirror surface to find the mirror surface normal. The sensor’s measuring axis was parallel to the DMI axis and the mirror surface normal. The projected angle θ shown in Fig. 6 can be calculated from these results and the direction of the projected beam.

3.3 Scaling Effect

A scaling factor is introduced if the sensor is tilted relative to the test surface. As seen from Eq. (1), the scaling factor can be derived from the triangulation sensor test geometry. The scaling sensitivity was tested by tilting the sensor relative to the double-sided flat as shown from Fig. 13.

In Fig. 13, θ is the beam projection angle, and φ is in the plane of rotation angle of the sensor. (The sensor readout is insensitive to the out-plane rotations for small angles.) This scaling effect needs to be taken into account during the SOC alignment to the test part. It has a sensitivity of ∼0.015∕ deg for the particular sensor we used. As the test surface aspheric departure gets larger, the scaling effect from the sensor angular alignment becomes more significant. For instance a test surface with a 2 mm peak-valley aspheric departure and with 0.01 deg alignment errors will have scaling induced errors up to 0.3 μm in P-V. This scaling sensitivity puts a long-term stability requirement on the SOC. The test triangulation sensor system is sensitive to the angular drift between the sensor and the test surface.

As seen from Eq. (1), the projection angle can also be checked by varying some known angles between the sensor and the double-sided mirror and then reading out the scaling factor from the difference between the sensor and the DMI readings.

3.4 Angular Dynamic Range

A large angular working range from the sensor is desired, as we would like the sensor to continue to work when the test surface normal is not parallel to the measurement axis of the sensor. This is the situation for the SOC when the surface has a large departure from a sphere or a flat. As seen from Fig. 9, we need to know the angular range over which the sensor reads out the surface sag d, independent of the slope of the surface. The triangulation sensor we are using has the property that the detector plane is conjugate with the testing location, which is along the line of the projected beam as shown in Fig. 4. Due to this imaging relationship, the sensor is insensitive to the local slope variations of the test surface.

A possible experiment layout for testing the sensor angular range based on the existing setup is shown in Fig. 14. However, the angular range of the DMI is only a few minutes of arc and cannot support the large angular range we would like to investigate.

Another approach might be using a wedged double-sided mirror. However, multiple samples would be needed for sampling the angular range we are interested in, and the sampling is not continuous.

Instead, we decided to test the angular range by measuring a ground optical surface with a known shape as described below. The results show that the sensor maintains good linearity over the angular range tested of about +?3 deg.

3.5 System Calibration

3.5.1 Calibration under a CMM with a ground spherical surface

The initial system test was done using a CMM as the scanning device before the sensor was verified for service on the SOC as shown in Fig. 15. The x and y linear stages of the CMM were used to scan and record the coordinates of the mirror under testing. (The coordinates were further corrected with ray tracing.) The CMM z axis was locked, and the surface sag was read out by the sensor. A spherical convex ground surface was measured for calibration. The radius of curvature and diameter of the sample mirror was chosen so that the 2 mm sensor working range and a 3 deg angular range could be checked by scanning the full aperture of the mirror.

The data were collected continuously along the x axis for different y values using the CMM. Figure 16 shows an example of the data obtained from the scans. Each line in the figure corresponds to a particular x scan with a fixed y value. This shows the sag value seen by the sensor relative to the angle between the sensor axis and the normal to the sphere being tested. The performance of the sensor for different working angles is checked by this sampling strategy.

The setup offered a simple way of collecting a large number of data points rather quickly. In addition, it was easy to calculate the theoretical sag of the sphere at each point to compare with the measured values. After removing the ideal spherical shape, the difference map in Fig. 17 shows an rms difference of 0.62 μm. The errors are dominated by the scanning errors from the CMM rails, the deflection of the sensor due to moments induced by the readout cable, and the ground surface roughness. No systematic errors due to sensor angular dynamic range issues were noticed.

3.5.2 Measuring an aspheric surface

To check the system performance, we attached the calibrated sensor to the SOC to guide the grinding of a 1-m diameter 80 μm P-V aspheric surface. Figure 18 shows the comparison of the measurements from the SOC and from a null interferometric test when the mirror was in its polishing stage. The results show the SOC and interferometer data agree with each other to 100 nm rms or better. This shows that the triangulation sensor is reliable for measuring both ground and polished surfaces.

4 Conclusion

The SOC is an important metrology technique for highly aspheric surface testing because of its versatility and high accuracy. It is configurable for measuring concave, convex, and plano surfaces. It can make in situ measurements, and its high-precision performance can rival full aperture interferometric tests. We have shown how the measurement range of the SOC can be extended to ground surfaces by using a calibrated laser triangulation sensor, and we have shown how to carry out that calibration. The experimental data show that the SOC equipped with a triangulation sensor can measure a test surface to a precision of better than 100 nm rms. This significantly improves the optical fabrication efficiency by extending precision metrology into the grinding cycle of the fabrication.

References

1. T. L. Zobrist et al., “Measurements of large optical surfaces with a laser tracker,” Proc. SPIE 7018, 70183U (2008).

2. T. Su et al., “Scanning long-wave optical test system: a new ground optical surface slope test system,” Proc. SPIE 8126, 81260E (2011).

3. P. Su et al., “Swing arm optical CMM for aspherics,” Proc. SPIE 7426, 74260J (2009).

4. P. Su et al., “Swing arm optical coordinate-measuring machine: modal estimation of systematic errors from dual probe shear measurements,” Opt. Eng. 51(4), 043604 (2012).

5. P. Su et al., “Swing arm Optical CMM: self calibration with dual probe shear test,” accepted to Proc. SPIE 8126, 81260W (2011).

6. Acuity laser measurement, “White light confocal displacement sensor,” 043604 (Nov 2010) http://doc.diytrade.com/docdvr/1391688/21942722/1308161289.pdf.

7. Keyence America, “High-speed, high-accuracy CCD laser displacement sensor,” Keyence America (2012) http://www.keyence.com/ products/measure/laser/lkg/lkg.php,
http://www.micro-epsilon.com/ displacement-position-sensors/laser-sensor/index.html.

8. N. Van Gestel et al., “A performance evaluation test for laser line scanners on CMMs,” Optic. Laser. Eng. 47, 336–342 (2009).

9. B. Muralikrishnan et al., “Dimensional metrology of bipolar fuel cell plates using laser spot triangulation probes,” Meas. Sci. Technol. 22, 075102 (2011).

10. J.-H. Wu et al., “The application of laser triangulation method on the blind guidance,” Proc. SPIE 8133, 81330V (2011).

11. R. G. Dorsch, G. Häusler, and J. M. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Optic. 33, 1306–1314 (1994).

12. D. MacKinnon et al., “Lateral resolution challenges for triangulationbased three dimensional imaging systems,” Opt. Eng. 51, 021111 (2012).

13. M. Daneshpanah and K. Harding, “Surface sensitivity reduction in laser triangulation sensors,” Proc. SPIE 8133, 81330O (2011).

14. S. V. Mikhlyaev, “High-precision triangulation sensing of mirror surface,” Proc. SPIE 4416, 400–403 (2001).

15. S. V. Mikhlyaev, “Influence of a tilt of mirror surface on the measurement accuracy of laser triangulation rangefinder,” J. Phys. Conf. 48, 739–744 (2006).

16. P. Su et al., “Maximum likelihood estimation as a general method of combining sub-aperture data for interferometric testing,” Proc. SPIE 6342, 63421X (2007).

17. P. Su, J. H. Burge, and R. E. Parks, “Application of maximum likelihood reconstruction of sub-aperture data for measurement of large flat mirrors,” Appl. Optic. 49(1), 21–31 (2010).

18. Renishaw, “RLE system overview,” (2012) http://www.renishaw.com/ en/rle-system-overview–6594.

19. R. E. Parks and W. P. Kuhn, “Optical alignment using the point source microscope,” Proc. SPIE 5877, 58770B (2005).

20. J. H. Burge et al., “Use of a commercial laser tracker for optical alignment,” Proc. SPIE 6676, 66760E (2007).

Calculation Of The Vertex Radius Of An Off Axis Parabolic Surface Using The Sag Measured With A Three Ball Spherometer

Written by csadmin on March 9, 2016. Posted in Optical Testing, Published Papers. No Comments on Calculation Of The Vertex Radius Of An Off Axis Parabolic Surface Using The Sag Measured With A Three Ball Spherometer

Abstract

We describe a method of calculating the vertex radius of an off-axis parabolic segment using a three ball spherometer to measure the sag. The vertex radius is found by solving a set of six simultaneous, non-linear equations for the three coordinates of one of the ball centers and the corresponding three coordinates of the point of tangency of the ball with the surface.

1. Introduction

Three ball spherometers are the 1” micrometers of an optical shop and have been used for over 100 years to measure the radii of spherical surfaces.^1,2 The radius of curvature of a spherical surface can be calculated from a simple, three term formula that includes the measured sagitta, or sag, the distance from the central micrometer or indicator and the center of the balls and the radius of the balls.^3,4 Until recently, in spite of the simplicity of the formula that includes a square in one term, spherometers were sold with tables that converted sag readings into radii because optical shop personnel often did not have the skills or time to solve the formula. With the advent of pocket calculators it is now easy to solve the sag equation.

The sag equation is rather simple because the spherometer balls touch the surface at normals to the surface, and for a sphere these go through the center of curvature of the sphere and through the centers of the balls on the spherometer. In the case of an aspheric surface this is not so; the ball centers still lie on the normals to the aspheric surface but the normals do not go through the center of curvature of the asphere. The solution to this problem requires the solution of six non-linear, simultaneous equations, again something not easily undertaken even by rather mathematically adept optical engineers. However, with the advent of equation solvers such as the “Solver” function in Microsoft Excel, solution of these simultaneous equations is straight forward and fast. This is the problem we will illustrate.

2. Measured surface sag (T)

The measured sag, T, is the difference between the spherometer indicator reading when the spherometer is sitting on a plane surface, Fig. 1 (left), and when the spherometer is resting on the aspheric surface to be measured (right). We have shown the case for a convex surface but identical conditions hold for a concave surface with the exception that the sign of the indicator movement is opposite.

3. Geometry of the three ball spherometer

Referring to Figure 2, the spherometer has three balls on the points of an equilateral triangle. D is the diameter of the circle through the centers of the balls, S is the length of the side of the triangle and the distance between ball centers, P is the height of equilateral triangle, Ω the distance from the center of the triangle to the balls and r is the radius of the balls. B10 is the center of the ball located on the meridional plane of the parabolic segment and the vertex of the parabola lies above this ball. B20 and B30 are ball centers located symmetrically with respect to the meridional plane. Relationships of the sides of the spherometer are given below.

4. Geometry of the spherometer resting on the parabolic surface

We use a Cartesian coordinate system where z is the axis of symmetry of the paraboloid and y lies in the meridian plane. The origin is coincident with the vertex of the parent parabolic surface. When the spherometer is first set on the parabolic segment we place ball B10 on the meridional plane with the tip of the indicator approximately centered on the segment. At first we know only that X_B10 = 0, Y_B10 is approximately P less than the center of the segment and Z_B10 is r higher along the normal to the surface than where the ball touches the surface, Z_F10 = Y_B10^2/(2R_v) where R_v is the vertex radius of the parabola.

Referring to Fig. 3 we see the spherometer in two states, the final one (subscript 1) sitting on the surface and an initial one (subscript 0) where the calculation is begun. We start by assuming the plane of the three spherometer balls is perpendicular to the normal to the surface at F10. This allows us to calculate a starting position for B20, and by shifting the plane of the ball centers downward by r, the starting position for F20.

Then we assume the spherometer is iteratively rotated down to the surface about the center of B10. The point of contact, F10, will not change nor will the dimensions of the spherometer. However, as the spherometer is rotated so the other two balls rest on the surface the calculated dimensions of the spherometer will change. Since the spherometer dimensions cannot change we have a method of solution by simultaneously solving the six equations below for the centers and points of tangency of the floating balls.

5. Calculation of points of tangency to the parabolic surface of the equilateral triangle base

The six unknown variables are

and they are found by solving six equations simultaneously. The six equations are

The relationship of the ball center to the point of tangency along the normal to the surface

where α, β and γ are the angles of the normals to the surface at F21. Finally the distances between the balls

Once the six unknowns are found it is straight forward to find the coordinates of the center of the spherometer and project the indicator down to the surface of the parabolic segment normal to the plane of the ball centers. In general, the tip of the indicator will not hit the center of the segment exactly and a new value will have to be used for Y_B10 and the calculation repeated until the tip is close enough to the desired location. Notice too that we have calculated the sag assuming the vertex radius is correct. If the measured sag is different from the desired, the R_v in eq. 5 must be changed until the results match the measured sag.

6. Summary

We have shown how the sag of an off-axis parabolic segment as measured with a three ball spherometer may be used to determine the vertex radius of the segment. While this is not as easy as for a spherical surface, the calculation is well within the capabilities of an optical engineer with access to iterative equation solving software.

7. References

1. Grubb, H., Nature, 34, 85 (1886)

2. Ritchey, G. W., Smithsonian Contributions to Knowledge, 34 (1904)

3. Horne, D. F., “Optical Production Technology”, p. 324, Crane, Russak & Co., Inc., New York 1972

4. Malacala, D. “Optical Shop Testing”, p. 818, 3rd ed., Wiley-Interscience, 2007.

Practical Alignment Using An Autostigmatic Microscope

Written by csadmin on March 8, 2016. Posted in Centering, Published Papers. No Comments on Practical Alignment Using An Autostigmatic Microscope

ABSTRACT

Auto-stigmatic microscopes (ASM) are useful for bringing centers of curvatures of lenses and mirrors to the centers of balls used as part of an alignment fixture. However, setting up the fixture to get the balls used for alignment in a straight line to represent the optical axis generally requires another piece of equipment. We show that within a practical range, the autocollimation mode of a modern ASM can be used to align balls to an axis with about the same precision as they could be aligned with an alignment telescope, or laser tracker. As a lead in to this topic, we discuss our meaning of alignment, the means of positioning optically important features such as centers of curvature and foci to the coordinates specified on assembly drawings. Finally, we show a method of using an ASM along with other tooling to align a toroidal mirror using its foci.

1. INTRODUCTION

1.1 Definition of alignment

In a paper on alignment we should start by defining alignment. Every optical system is built to a design. First, there is the optical design, and it is used by a mechanical engineer to design a barrel or optical bench into which the lenses and mirrors are held. Throughout this design effort, the optical elements are assumed precisely located where the design specifies. It isn’t until the optical bench and elements come together that the issue of alignment comes up. It is at the assembly stage where the elements must be located precisely where the design or drawing specifies. This is alignment, getting optical features such as centers of curvature, axes, and foci positioned precisely as the design or drawing indicates.

In order to get the optical features located precisely, an instrument is needed that can sense the optical feature in question precisely. The same instrument, in conjunction with mechanical tooling, must also sense mechanical datums in the lens barrel or optical bench since the axes and foci of the optical system must be positioned precisely relative to specific mechanical features called out in the design. For example, the design may call for the optical axis of a system to be concentric with the outside of the cylindrical barrel of the assembly. Thus there must be a way of locating both the optical axis of the optical elements and the mechanical axis of the cylindrical barrel to assure coincidence of the axes.

1.2 Autostigmatic microscope

An instrument that can define both optical and mechanical datums in conjunction with some simple mechanical tooling is an autostigmatic microscope1 . In simplest terms, an autostigmatic microscope is a microscope that has a point source of illumination that is conjugate with a set of crosshairs in the image plane, either real physical crosshairs in an eyepiece or electronic crosshairs on a video display. If the autostigmatic microscope is perfectly aligned to the center of curvature of a sphere, the reflection from the sphere will form a well focused point image centered on the crosshair, that is, it defines 3 degrees of freedom (DOF). If the microscope is not well aligned the spot will be out of focus and shifted laterally from the crosshairs. Fig. 1 shows the typical optics of an autostigmatic microscope both aligned and misaligned with a concave sphere.

In order to make the autostigmatic microscope more useful, the addition of full field, or Kohler, illumination is helpful so the microscope can image as an ordinary microscope does. If the focal plane of the imaging arm and the autostigmatic focus are the same, the instrument is even more useful in that both types of illumination can be used simultaneously as will be illustrated in the first example of its use in section 2. A microscope with these features that is commercially available is called a Point Source Microscope2 and the optical paths in the microscope are shown in Fig. 2.

2. ALIGNMENT OF A STAGE COINCIDENT WITH AN AXIS OF ROTATION

2.1 Some definitions

As a first example of alignment with an autostigmatic microscope, we will consider the alignment of a single axis linear stage with the axis of a rotary bearing. The specific example will be a vertical stage over a rotary table used for centering lenses, but the example is just as applicable to the alignment of a lathe headstock with the lathe bed and tailstock center.

As with the term alignment, it is best to start with a definition or two. Since we are aligning an axis we should note that an axis is a line and a line is defined by two points, or a point and an angle in two planes, in other words by four DOF. Rotation about the axis and distance along the axis are not defined.

Another term used in conjunction with axes of rotation is centering and this can be confusing since there can be two meanings to centered. For a point, or spot, to be centered on an axis of rotation it is stationary when the axis is rotated. For the axis to be centered on the microscope, the axis must lie on the microscope axis, or crosshairs. The illustration in Fig. 3 helps explain the difference in the two concepts assuming a rotating table.

In the upper left of Fig. 3 the spots of light produced by the autostigmatic microscope move in a circle as the rotary table revolves about its axis. This means the light spot does not lie on the axis of rotation. In the lower left, the light spot is centered on the axis of rotation, but the axis of rotation is not centered on the microscope axis, or crosshairs. In contrast on the upper right, the spots are rotating about the microscope axis but are not centered on the axis of rotation, while in the lower right, the spots are both centered on the axis of rotation and that axis is centered on the microscope axis or crosshairs. In short, if the spots are moving in a circle, the spots are not centered on the axis of rotation meaning the center of curvature from which the spots are reflecting does not lie on the axis of rotation of the rotary table.

The same concept applies to centering an object in the microscope field of view. Consider the crosshair target in Fig. 4 that is imaged by the microscope. One end of the crosshair is indicated by a circle to show that the crosshair rotates as its center traces out a circle about the axis of rotation of the rotary table.

As seen in the lower left view of Fig. 4, the crosshair will rotate as the axis is rotated but the center of the crosshair remains stationary in the field of the microscope indicating that the crosshair lies on the axis of rotation but is not centered on the axis of the microscope while it is in the lower right view. The two types of centering require two physically different operations; to center the spots or crosshair on the axis of rotation, the optical surface must be translated or tilted relative to the axis of rotation. To get the axis of rotation to lie on the microscope axis either the microscope must be moved to the axis, or the axis of rotation translated or tilted to make it coincident with the microscope axis or crosshair.

2.2 Alignment of an axis

Now to the alignment of the vertical stage to the axis of the rotary table. Fig. 5 illustrates the situation. There is a rotary table in the lower part of the Figure and a vertical stage to the right. The Point Source Microscope (PSM) is shown in two positions on the stage, one focused on a target on the table using the imaging light source and one at the center of curvature of the plano concave lens lying on the table. These two foci of the PSM are the two points on the line defining the axis of the stage since the PSM is physically tied to the stage. The axis of the table is virtual because it cannot been seen, but it is made real by use of two simple fixtures, a crosshair target on the rotating table defining one point on the axis and the plano convex lens defining the other.

To align the axis of the rotary table to the axis of the stage as defined by the PSM focus, we must both make the target crosshair and the reflection from the concave surface stand still in space. This is accomplished by translating the target and lens, separately, relative to the table top. The target can be a piece of paper with the lens sitting on top. When the PSM in the imaging mode is focused on the paper the situation will typically look like the case in the upper left of Fig. 4. The target must be decentered relative to the rotary table top until it looks like the case in the lower left of Fig. 4. Then the target crosshair lies on the axis on the rotary table.

Similarly, at the center of curvature of the lens, the situation will look like the case in the upper left of Fig. 3 when the PSM is focused at the center of curvature using the autostigmatic mode. If the vertical stage is very decentered, the stage may have to be translated to get the return image in the field of view of the microscope, but the real task is to make the reflected spot cease to move as the table is rotated, that is, to achieve the case shown in the lower left of Fig. 3.

Once the crosshair and spot are centered, that is, not moving except in the case of the cross, rotating about the center of the cross, it is then necessary to translate and tilt the stage until the cross and spot both lie on the crosshairs in the microscope. When the alignment is finished the crosshair target will look like the case in the lower right of Fig. 4 and the spot will look like the lower right of Fig. 3. It is important to make these adjustments in the correct order, center the targets on the rotary axis first and then adjust the column. Further, remember that the target errors are seen doubled due to the rotation, and in the case of the spot, reflection. Do not try to make the full correction all at once. You will overshoot every time and the alignment will take at least twice the time it should. Make the adjustments in fractions of what it looks like the translation needs to be.

Notice, too, this alignment could not be done without some simple fixturing to “realize” datums that are virtual without the fixturing. In general, the most useful fixtures are spherical surfaces, physical realizations of points, and next most cylinders, realizations of lines. These make the most precise realizations because if lens surfaces or high grade spherical balls are used, their centers can be located to <1 μm. With images of crosshairs, it is difficult to use centroiding algorithms to achieve micron centering, but 10 μm centering is easy with images.

3. ESTABLISHING AN AXIS

In this example, we will describe how to use an infinity conjugate autostigmatic microscope to establish an axis. This is important because the majority of optical systems have a single axis of symmetry and alignment for these systems means having the centers of curvatures all lying on a common axis. In passing, the easiest way of establishing an axis is to use a good rotary bearing like in the previous example. However, there are many times when such a table is not available or the optics are too large to put on a rotary table.

For the alignment of a system with a single axis, fixtures or datums have to be positioned at precise locations along the axis. The lateral positions are tighter than the axial locations, but the axial locations still need to be carefully controlled. The most common way to do this alignment is with an alignment telescope3 , but it is a tedious proposition and almost always requires two people, one looking in the eyepiece and one making the adjustments to the fixtures.

There is an alternative that works well over distances from about 0.5 m to 3 m. That is, remove the objective from the autostigmatic microscope and use the collimated beam projected by the 6 mm aperture. When the collimated beam reflects off a spherical surface, the return wavefront is nearly spherical, and for that part of the wavefront returning through the aperture on the microscope it is perfectly spherical assuming a good grade spherical ball. This situation is like that shown in Fig. 6 where a steel ball is used to maximize the reflected light.

The spherical wavefront reflected off the ball will have a radius of curvature of approximately the distance from the microscope to the ball. Assume for this example the distance is 1 m and the ball is 25 mm in diameter. The PSM has a 6 mm aperture and a 100 mm tube lens. This means the return wavefront will produce a spot about 0.6 mm on the 1/3” CCD detector that is 3.6 mm in the short direction. While this is not a perfectly focused spot, the centroiding algorithm in the PSM software works just as well on out of focus images as well focused ones. The limit here is that the entire spot lie on the CCD. In this example, the shortest distance from ball to PSM would be 250 mm to keep the image from being bigger than the detector. If a shorter distance were needed, the 6 mm aperture could be stopped down. Also, short distances mean plenty of light gets back through the aperture stop.

The far distance is limited by the light available but has the advantage that the spot on the CCD is smaller. For the example, the 6 mm collimated beam is spread into a cone of 0.235 steradians. At a 1 m distance, the 6 mm aperture subtends just 36 μsteradians, or collects just 1.5×10-4 of the reflected light assuming a top hat intensity distribution. On the other hand, the PSM has an intensity dynamic range of about 1×105 . Thus a dynamic range at a distance of 3 m or more is perfectly reasonable.

Figure 7 shows the PSM used in the autocollimator mode to align a set of balls on a common axis. Here the farthest ball is aligned first. A post such as used to position lens mounts can be used as a kinematic locator for the ball. The post is on an xyz stage and the stage is adjusted until the reflection from the ball is centered on the PSM display. Subsequent balls are similarly adjusted. The axial spacing is set using an inside micrometer.

There are several advantages over using an alignment telescope in addition to those already mentioned. The collimated beam is bright enough to see under ambient lighting so it is easy to set up the initial alignment, the balls are easier to use than crosshair targets needed with an alignment telescope, the set up has no moving parts in the alignment instrument and there is nothing to focus. Finally, the centroiding is completely objective.

In terms of precision, the centroiding algorithm has a sensitivity of just under 1 arc second, or ~5 μradians. Even with out of focus images on the detector experience has shown this to be the case. The means that balls aligned using this method can be positioned to about ±5 microns at the distance of 1 m. Over the range where sufficient light reaches the detector this sort of precise is on the order of that achievable with a laser tracker.

4. ALIGNMENT OF SYNCHROTRON MIRRORS

The final example is one that I have not had the opportunity to try out but is illustrative of a technique that could be applied to many situations where high precision alignment is required in a multi-dimensional space. Further, it shows how the autostigmatic microscope can be used along with other sophisticated optical tooling such as a laser tracker4 to increase the value of both the tracker and the microscope. The whole example is laid out in Fig. 8 and the various steps will be described in the text, one after the other. This is again an example of how there is a particular sequence of steps that must be gone through to achieve the final alignment.

The synchrotron mirror is⁵ almost plane and thus has the same DOF as a plane mirror, rotation about the axes parallel to its face and translation perpendicular to its face. Using these 3 DOF the mirror is adjusted so the light from the fiber tip is focused at the PSM focus. The focus will be a line normal to the page and have a width limited by diffraction set by the apparent width of the mirror looking into it at grazing incidence in the plane of the page. The other 3 DOF are set using mechanical datums on the mirror and the structure holding the mirror.

In this example we have tried to demonstrate the use of the autostigmatic microscope with other optical tooling used in large optical projects. The SMR’s used by the laser tracker are completely compatible with the use of steel balls used in other autostigmatic microscope applications. The imaging aspect of the PSM permits its use in the location and centering of an optical fiber tip that will then be used as a point light source at one conjugate of the system. Finally, the PSM can be used to detect the focused image coming single pass through the optical system to align the synchrotron mirror.

5. CONCLUSIONS

We have first set out a working definition of optical alignment as determining the location of an optical, as opposed to a mechanical, feature, and then moving that feature to the location specified in the design of the optical system. Then we have given a simple example of the alignment of a center of curvature to an axis of rotation to show the difference between alignment of the feature to the axis of rotation, and the alignment of the axis to the instrument making the measurement. This is followed by an example of using an infinite conjugate autostigmatic microscope to align a series of spherical ball targets in a straight line. Finally, we show how to use an autostigmatic microscope in conjunction with a laser tracker to systematically align a synchrotron mirror using the mirror’s two foci.

REFERENCES

[1] Steel, W. H., “The Autostigmatic Microscope”, Optics and Lasers in Engineering 4(4), 217-227 (1983).
[2] www.optiper.com
[3] For example, http://www.brunson.us/p/AboutAlignTel.asp
[4] For example, http://www.faro.com/laser-tracker
[5] http://www.bnl.gov/nsls2/project/CDR/Ch_08_Radiation_Sources.pdf, p. 18.

Lens Centering Using The Point Source Microscope

Written by csadmin on March 7, 2016. Posted in Centering, Published Papers. No Comments on Lens Centering Using The Point Source Microscope

ABSTRACT

Precision lens centering is necessary to obtain the maximum performance from a centered lens system. A technique to achieve precision centering is presented that incorporates the simultaneous viewing through the upper lens surface of the centers of curvature of each element as it is assembled in a lens barrel. This permits the alignment of the optical axis of each element on the axis of a precision rotary table which is taken as the axis of the assembly.

1. INTRODUCTION

Lens centering is a crucial step in the manufacture of rotationally symmetric optical systems. For optical systems to deliver maximum performance the optical design must be superior, the lens elements accurately manufactured and the elements well centered in a barrel. With experienced designers, sophisticated software and vast computing power, superior designs are easier to produce than in the not too distant past. With phase shifting interferometers and mature computer controlled polishing it is possible to produce optical surfaces accurate to a few nm rms or better.

In order to take full advantage of an excellent optical design and well manufactured optical surfaces the lens elements must be well centered in the assembly. Without a careful and accurate job of centering the money put into design and polishing will be lost. This paper discusses how to do a superior job of lens centering by simultaneously sensing both centers of curvature of each element from above as the element is placed in the lens barrel and the barrel is rotated about its axis. Each element is adjusted in tilt and decenter until there is no motion in either center of curvature.

To demonstrate this procedure we will first discuss the definition of centering and give a specific lens design as an example. Then we will show how to locate the centers of curvature of the lens surfaces looking from one side of the lens only. The instrument used to optically locate the centers of curvature, an autostigmatic microscope that we have called the Point Source Microscope (PSM), will be described and an example of how it would be used to find the center of curvature for each lens surface will be given.

Finally we discuss how the tilt and decenter of the surfaces affect performance of the lens system and how sensitive the PSM is to these centering errors to give a feel of the performance improvement using this superior centering technique. This leads directly into the conclusions.

2. DEFINITION OF CENTERING

2.1 Optical axis of a lens element

The optical axis of a lens is the line between the centers of curvature of the two surfaces. It is a paraxial property that is seldom given any thought but is the basis for this discussion. Aspheric surfaces do not change the definition. If the whole aspheric surface produces too much spherical aberration to center on the reflected return image the surface may be stopped down to effectively make the surface paraxial.

There are several things to note about this definition; it is completely independent of the mechanical features of the lens such as the periphery or seat, and the definition is incomplete without considering both lens surfaces because a single sphere has no intrinsic axis. If a lens is poorly centered its periphery is not concentric with the optical axis nor is its seat perpendicular to the axis, but it still has a well defined optical axis.

In terms of the larger aspects of a lens system, if both object and image lie on the optical axis of the lens system it is being used on-axis. If one or more elements within the lens system are decentered it becomes difficult to define the axis of the system. Probably the best definition then is the angle the lens must be tilted to produce the best image, that is, how must the lens be tilted about the line joining object and image to produce the best image.

2.2 An example lens system

To demonstrate the ideas about centering we give the design of a three element, all BK7, f/1 infinite conjugate lens with one aspheric surface. The design is shown in Fig. 1 and the parameters given in Table 1. The design has about 0.035 waves P-V of spherical aberration and is 0.0075 waves rms at the design wavelength of 0.55 µm.

3. LOCATING THE CENTERS OF CURVATURE OF THE LENS SURFACES

Before looking at the centers of curvature of the lenses we show the three lens elements in their cell (see Fig. 2) to give a feel for the physical constraints on centering. The cell is sitting on top of an air bearing rotary table and has been adjusted so there is zero runout or decenter and no tilt, that is, the axis of the cell is coincident with the axis of the rotary table. It is further assumed there is no error in the table bearing, an assumption good these days to 50 nm sorts of dimensions.

It is clear from Fig. 2 that all optical sensing of the centers of curvature of the elements must be done from above the lens system. While rotary tables are available with through holes1 and use of the underside of the lens may make some examples easier, particularly in volume production, we show there is no need to view the lens from the bottom.

It is clear how to locate the centers of curvature of the upper sides of the lenses, the centers of curvature lie a distance equal to the radius of the upper surface below the lens element in question. A positive lens with a long working distance, or back focal length, will have to be placed above the example lens system. As a reasonable choice a 100 mm efl lens could be used because the longest convex radius is 94.286 mm on the upper surface of Element 3.

Now we will use this positive lens along with a lens design program such as Zemax2 to find the apparent location of the center of curvature of the rear surface of each lens element looking through the upper surface. For Element 1, the first one that will have to be inserted in the cell, the Lens Data Editor is set up as in Fig. 3.

The object is set at infinity feeding the paraxial (perfect) 100 mm efl lens that is surface 1. The distance (thickness) between the paraxial lens and the front surface (line 2) of Element 1 is the unknown (or variable) we must find. On that line (2, also the stop in the system) is the radius of the upper surface, 26.172 mm, the thickness of the lens, 5 mm, and the material, BK7. The next line (3) is the rear surface of Element 1 with a radius of 33.70628 mm. If the light is to strike that surface at normal incidence, the condition necessary when a point source is at the center of curvature, the light must come to focus a distance equal to the radius of the surface farther to the right. That is why the thickness for the rear surface is also 33.70628 mm as is assured by using a pickup (P) from the radius column.

The next, or image, line shows the light is focused at this distance because the diameter of the image is very small when the thickness from the paraxial lens to the front surface is 48.627 mm, a value found by using the design optimizer.

Notice the thicknesses do not add to 100 mm showing the effect of refraction at the front surface. Another way of finding the distance between the center of curvature and the front surface of the lens is to use paraxial ray tracing to find that

Notice this is just the paraxial lens focal length minus the paraxial lens to front surface thickness of 48.627, or 51.373. The 5 µm difference between the results can be accounted for by not using a small enough aperture size to make the lens design optimizer give a truly paraxial result.

Fig. 4 shows the results of using this same technique on all three lens elements in the system. Notice for Element 1 that the distance from the rear surface to the focus, 46.373, plus the lens thickness, 5, and the distance from the lens front surface to the paraxial lens, 48.627, add to 100 mm, the efl of the paraxial lens. Also shown are the physical centers of curvature of the two surfaces. The same logic holds for the other two elements but in these cases the center of curvatures of the rear surfaces after refraction are above the front surface by 556.182 and 106.416 mm for Elements 2 and 3, respectively.

Although the example lens system does not have a concave front surface it is obvious that the center of curvature of this surface would be above the surface by the radius of the front surface. The method for finding the location of the center of curvature of the rear surface after refraction in the front is the same as for this example. Now that we have shown how to locate the centers of curvature of both lens surfaces from above the lenses we show how to use the Point Source Microscope (PSM) to view these locations.

4. POINT SOURCE MICROSCOPE (PSM)

The PSM is a video metallographic, or reflected light, microscope using Köhler illumination to provide uniform intensity over the field of view. In addition, the PSM has a point source of illumination produced by the end of a single mode fiber pigtailed to a laser diode that is conjugate to the microscope object plane as shown in Fig. 5 below. This point source of light makes the PSM into an autostigmatic microscope, and it is this feature that is used to view the centers of curvature of lens elements during centering.

When the point light source is used and the focus of the microscope objective is conjugate with the center of curvature of a lens element or mirror surface the light will strike the surface at normal incidence and be reflected back to the microscope objective focus and on to the CCD detector where it can be viewed on a monitor. When the alignment with the center of curvature is precise in lateral position and focus the return spot will be a well focused and centered on the detector. If the alignment and focus are less than ideal the spot will be defocused and decentered as shown in Fig. 6.

It remains to show how two PSMs can be set up to view both sides of a lens element simultaneously so the optical axis can be established and aligned. It is also obvious that there are limits as to how well lenses need to be centered and the PSM can be used to tell if the centering is good enough by measuring the excursion of the return spot of light as the rotary table is turned.

5. VIEWING BOTH CENTERS OF CURVATURE SIMULTANEOUSLY

5.1 Scheme for simultaneous viewing

Because the optical axis of a lens element is the line joining the centers of curvature and it is that axis that must be aligned to both the cell axis and the other elements within the lens system it is most expedient to view both centers of curvature simultaneously during centering. Although the centering operation can be done be viewing just one center at a time it is very tedious to do it this way because the alignment of one center may, and usually does, disturb the centering of the other.

Going back to our example lens we look at Element 1 first. In Fig. 7 we show just the location of the upper or front surface center of curvature and the rear surface center after refraction in the front surface because these are the two centers we see with an autostigmatic microscope

Now instead of using a 100 mm efl lens at infinite conjugates we use a 50 mm efl lens at 1:1 finite conjugates to relay the front surface center of curvature to the right side of the 50 mm lens. However, this same lens can be used to relay the rear surface apparent center of curvature with an object conjugate of 125.20 mm to the right side with an image conjugate of 83.254 mm accomplishing just what we wanted to do, image both centers of curvature simultaneously. It now remains to separate the centers so they can each be viewed without having to move a PSM.

This is most easily accomplished with a minor modification to the layout in Fig. 7, the introduction of a beamsplitter with a 50 mm efl lens cemented to it so there is only one optical component above the lens element being centered. The setup will then look like that in Fig. 8 where the optical paths of both PSM’s are shown. We have neglected the slight axial shift in the conjugates due to the optical thickness of the beamsplitter as well as the small introduction of spherical aberration it will produce. In practice the f/number of the cone of light used for centering is so narrow that the exact axial position and small amount of aberration is not critical. Further, all the conjugate calculations have been first order, or paraxial, so they are not exact but close and quite good enough for real implementation because what we are ultimately interested in is whether the relayed return spots from the centers of curvature move or not as the rotary table is turned.

5.2 Modification for viewing with two autostigmatic microscopes

Fig. 8 is rather busy but it shows the entire setup for centering Element 1. The two PSM’s send out cones of light that somewhat over fill the beamsplitter/relay lens combination and the relay lens serves as the stop as it sends converging beams of light to both the center of curvature of the front surface and to the center of curvature of the rear surface after the light has been refracted in the front surface. The light reflects off the front and rear surfaces of the lens and retraces itself back into the two microscope objectives. Both reflected light beams enter both objectives but the beams from the incorrect conjugates are so far out of focus that very little light reaches the detector and is inconsequential. Only light from the correct conjugate is well focused and bright on the detector of that PSM.

Similar setups work for the other two lens elements even though the rear surface centers of curvature appear above the front surface. Fig. 9 shows the conjugates for these two elements using the same 50 mm efl finite conjugate lens. In this example it just turned out that the same focal length lens could be used for all three elements. In general this will not be the case but a relatively few beamsplitter/relay lens combinations can be used for practically any lens element that needs centering. In fact, the relay lens is only 5.714 mm in from of Element 3 in Fig. 9, a distance that may be too close for comfort. A 60 or 75 mm efl relay lens could have been used just as well without much increase in the total dimensions of the measurement setup. Obviously all the relay lens conjugates would differ from the example.

It is further clear that the relay does not have to be used at 1:1 conjugates in one of the cases. The options for finding suitable conjugates is almost limitless, something that is a possible drawback in finding good conjugate solutions. However, it does show the flexibility in this system of simultaneous conjugate viewing. Along the same line the PSM does not have to be used with expensive microscope objectives since this application does not call for full field imaging. Inexpensive, longer focal length doublets can be mounted in lens tubes and mounted to the PSM using objective adapters. The beamsplitter would still have to be used but all the optical power would be in the PSM itself.

6. SENSITIVITY OF CENTERING

6.1 Theoretical considerations

Now that we have seen how to view both centers of curvature simultaneously we need to ask how sensitive are the PSM’s to tilts and decenters of the three lens elements in this example. For the front surfaces this is easy; if the surface is decentered 1 µm, the return spot will move by twice that because of the doubling on reflection. When the lens is rotated 180º with the rotary table the lens is decentered in the opposite direction so the total spot motion is 4 µm for a 1 µm decenter from the rotary table axis of rotation. Depending on the objective used, and the 5x would be typical for centering operations, there is another factor of 2.5 so the image motion at the detector is 10 µm. The detector has 4.65 µm pixels and can centroid to 0.1 pixels. Working backwards gives us a theoretical sensitivity to front surface decenter of about 47 nm.

Regarding rear surface decenters, clearly a front surface decenter will affect the apparent decenter of the rear surface. Rather than do a lot of calculations the pragmatic approach is to center the front surface first and then the rear surface decentered will not be affected except by a scale factor.

For tilts the sensitivity is linearly proportional the radius of curvature (or apparent radius) of the surface in question. Working backwards from the above example we find the sensitivity to tilt about 10 seconds of arc per mm of radius. Thus if a concave surface had a radius of 10 mm the PSM would be sensitive to a 1 second tilt of the surface and to a 0.1 second tilt if the radius of the surface were 100 mm.

It may be asked what is the effect of a tilt or decenter on the lens performance. A tilt displaces the chief ray so the lenses following the tilted element are in effect decentered. A decentered lens deviates the chief ray from the optical axis making the lenses following appear to be tilted. This is nicely shown in a recent paper by Burge³.

6.1 Practical considerations

Now it is time to look at the real situation in most instances in terms of lens edging and assembly in a cell. We will first look at assembly to gain an insight as to what we need to look at in terms of edging tolerances. When a lens is set in a cell the lower spherical surface seats on a step shaped land that was machined into the cell that is concentric with the axis of the cell in the radial direction and perpendicular to the axis in the axial direction as shown in Fig. 10. In a perfect world where there are no burrs or contamination on the seat this arrangement assures that the center of curvature of the lower surface lies on the axis of the cell.

The upper surface center of curvature will not lie on the axis of the cell if the lens was poorly edged, and in general, will not lie on the axis anyway. The optical axis of the lens is shown on the left of Fig. 10 and shows the decenter of the upper surface. This is easily corrected by decentering the lens to bring the upper surface center of curvature onto the axis of the cell. The lower surface remains centered because the edge of the seat is equidistant from the center of curvature of the lower surface. Clearly the cell bore must be sufficiently oversize to permit the decentering of the lens.

This brings us to tolerancing for edging in the case where lenses will be centered in a cell during assembly. Many lens design programs have automatic tolerancing schemes built in and they vary every possible lens parameter to see the effect on performance. Single surface tilts help with tolerancing the seats in the cell but have nothing to do with the lens element as a whole. Further, since the rear surface of the lens always (in a perfect world) stays aligned the tilt and decenter of the upper surface are one and the same; removing the decenter removes the tilt and vice versa. This means the lens element does not have to be toleranced for both.

Ultimately the consideration will come down to how much oversize does the bore have to be to insert the element without it getting hung up in the bore and how much clearance must be allowed for thermal changes between glass and cell, and what is the minimum practical thickness for a shim to center the lens in the cell. The answer to shim thickness is about 10 µm and numbers comparable to this for the other considerations for lenses in the 25 to 50 mm diameter range. It is such considerations that help with the decision to tolerances and not try to center during assembly or looser edging tolerances and center during assembly. This also becomes a question of how many units will be produced and the diameter of the elements. Clearly the shorter the focal lengths and smaller the lens diameters the more critical the decentering in order to keep the ratio of decenter to focal length a small number.

For those who like to go through the tolerancing numbers without the obscuration of the software the reader is referred to an excellent book by Gerrard and Burch4 on matrix methods in optics.

7. CONCLUSIONS

With modern lens design, interferometric testing of optical surfaces and computer controlled polishing it is possible to produce suburb optical elements. All the potential gain in performance due to these factors can be lost at the last step of assembly unless there is a method of assuring the lens elements are well centered in their lens cell.

We have showed how to implement such a method to assure the centering as lenses are assembled using a precise rotary table and a modern version of the classical autostigmatic microscope. The method can sense the centers of curvature of both surfaces of an element so that any errors in tilt and decenter can be corrected during assembly to the sub-micron and sub-second level.

The device that makes this possible is a classical alignment instrument, the autostigmatic microscope fitted with a bright point source of light, a sensitive digital video camera and monitor to allow convenient viewing of the return image and software to process the video image to give metrics of the degree of centering.

In addition to its application in lens centering the Point Source Microscope is being used for the alignment of cell phone lenses, astronomical telescopes, and terahertz optics. Metrics are returned at frame rates so the effects of adjusting alignment can be viewed in real time. In some instances users are finding the PSM is more convenient and easier to use than more expensive interferometric test methods.

REFERENCES

1. J. Heinisch, et. al., “Novel Technique for Measurement of Centration Errors of Complex, Completely Mounted Multi- Element Objective Lenses”, Proc. SPIE, 6288, 628810, (2006).

2. Zemax Development Corp., www.zemax.com.

3. J. Burge, “An easy way to relate optical element motion to system pointing stability”, Proc. SPIE, 6288, 628801, (2006).

4. A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics, particularly Appendix B, Dover Publications (1994).

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Using The Point Source Microscope (PSM) To Find Conjugates Of Parabolic And Elliptical Off-axis Mirrors

ABSTRACT

1. INTRODUCTION

2. GEOMETRY OF AN OFF-AXIS PARABOLA

3. GEOMETRY OF AN OFF-AXIS ELLIPSE

4. MEASURING THE SAGITTAL AND TANGENTIAL RADII

5. USE OF THE SAGITTAL AND TANGENTIAL FOCI FOR ALIGNMENT

6. RESULTS OF MEASUREMENTS ON AN OFF-AXIS MIRRORS

7. CONCLUSION

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1. Introduction

2. Measured surface sag (T)

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ABSTRACT

1. INTRODUCTION

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ABSTRACT

1. INTRODUCTION

2. DEFINITION OF CENTERING

3. LOCATING THE CENTERS OF CURVATURE OF THE LENS SURFACES

4. POINT SOURCE MICROSCOPE (PSM)

5. VIEWING BOTH CENTERS OF CURVATURE SIMULTANEOUSLY

6. SENSITIVITY OF CENTERING

7. CONCLUSIONS

REFERENCES

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