Robert E. Parks Optical Perspectives Group, LLC Tucson, AZ 85750
Daewook Kim J. C. Wyant College of Optical Sciences The University of Arizona Tucson, AZ 75721
INTRODUCTION
Following the discovery of so called non- diffracting Bessel beams [1], they have been used for a number of exotic purposes such as trapping single atoms and aiding in the discovery of exoplanets. We discuss more mundane but practical methods applicable to precision engineering, and the physical ray tracing of a ball lens in transmission to determine if it behaves as geometrical optics predicts.
First, we discuss the definition of a Bessel beam (BB) and how one is created [2]. Then we look at some of the BB properties to give the context of why they are valuable tools for tracing rays in the laboratory as opposed to mathematical ray tracing on a computer (i.e., non-realizable in practice due to the divergence of a very thin ray). We end by showing experimental results from using a BB to trace rays through a ball lens to show how well the experimental results match the computer paraxial ray trace of the same lens.
BESSEL BEAMS
A BB is simply a light intensity pattern in a plane perpendicular to its axis of propagation that is described by the Bessel function, (J0)2. It has a bright central peak surrounded by rings of decreasing intensity. The beam core is non- diffracting in the sense that its diameter is much less than you can create with a collimated beam by any other means. The core of the beam is 100 to 1000 times smaller in diameter than the beam from a HeNe laser, for example.
In addition to being much smaller in diameter (i.e., spatially well-localized) than a laser beam, BBs are robust and less affected by air turbulance because they propagate through a larger volume of air which helps to average out the environmental effects [3].
One way of creating a BB is shining a collimated or spherical wavefront on a plane grating made of uniformly spaced concentric circles [4]. In practice today, such a grating is made by ebeam lithography on a photomask substrate with circle spacings on the order of 10 μm. It is particularly convenient to illuminate the grating with a point source of light made by the end of a single mode optical fiber pigtailed to a laser diode. The advantage over a collimated wavefront is that the length of the BB is much longer than with collimated illumination [5] and you do not need a good collimating lens the size of the grating.
A disadvantage of point source illumination is that the central core of the BB expands as the beam propagates but the core maintains a peak intensity about 10 times that of first and subsequent surrounding rings. This means that the simplest sort of centroiding algorithm can still find the location of the core precisely.
PHYSICAL RAY TRACING
We know from simple geometry and lens design software that when a paraxial ray enters a lens at a known ray height and angle that it exits at the same height but different angle at the principle plane. In 1996, Santarsiero showed theoretically that BB behaved as paraxial rays propagating thorough ABCD optical systems [6]. We show experimentally that when a BB is projected through a ball lens, the beam traverses the ball as though it was a single paraxial ray in a lens design program.
Obviously, how the rays exits a lens will also depend on the intial alignment of the lens to a coordinate system. To avoid questions of alignment, we demonstrate physical ray tracing using a ball lens because such a lens cannot by misaligned in tilt. The only data of a ball lens are its radius and physical center; its center of curvature. If the detector of the position of the BB is centered on the BB prior to inserting the ball lens in the beam and the ball lens is centered so the beam is again centered on the detector, the BB must be passing through the center of the lens and the lens is free of tilt since a sphere has no axis [7].
We show that the BB traverses the lens and continues to propagate in free space precisely as predicted by geometrical optics, and that the beam’s height and angle can be measured at any arbitrary distance along the axis of the lens as the BB is nicely localized in space with a well defined central peak intensity.
PRACTICALITIES OF USING A BESSEL BEAM
Before describing the experimental results it is sensible to ask if there are any benefits of using BBs for aligning optics over classical methods of simply locating centers of curvature and foci. We believe there several reasons. First, it is often impossible to optically reach some centers of curvature either because there is mechanical interference or the center of curvature is inside a lens system so far that no practical long working distance objective can reach it.
Second, because the BB height can be located any distance from a lens just as you can find a ray location any place you insert a dummy surface in a lens design program, you can position the sensor far from the lens so that a small displacement of the beam gives great angular sensitivity. Another advantage of moving the sensor away from the lens is that allows plenty of free room around the lens for work on making adjustments to the lens.
Third, by using a BB as a reference axis there is no need of a rotary table for centering. If the sensor is aligned with the beam prior to inserting a lens, the lens is not centered until the beam is again centered on the sensor. Further, it is much faster and less tedious to center with simple x, y motion than having to rotate the lens a full revolution between each adjustment to see if the adjustment was correct. With the BB you have immediate hand/eye feedback.
EXPERIMENTAL SETUP
Our experimental setup consists of a BB generator module on a translation stage to move the BB across the ball lens. We use a ball because it never need correction for tilt. The ball sits in a circular seat that acts as a kinematic mount so the ball is easily removed to check on the incident BB location. Above the ball is a video microscope, a Point Source Microscope [7]in this case, to sense the BB location. Well focused images are only seen in the objective focal plane so we know the axial height at which the BB is located precisely as illustrated in the schematic of the setup in Fig. 1a. Figure 1b shows a photo of the ball on its mount under the microscope objective.
Fig. 1a. Schematic of the experimental setup
Fig. 1b. Setup showing ball in kinematic seat
As a vertical zero reference we use the Cat’s eye reflection from the upper pole of the ball and measure positive distances above the ball. For negative distances the objective is focused in the ball. The optical focal plane will be different from the physical focal plane because of refraction. Our measured vertical distances refer to the physical height of the focal plane.
Because the field of view of the 10x objective and camera are limited to about 1 mm we set an upper limit of the BB motion as ± 400 μm and viewed the BB as it moved through the field of view as in Fig. 2. We also wanted the experiment to simulate paraxial conditions as much as possible.
The stage moved at a constant velocity from 400 to – 400 μm while the PSM software logged the spot position against a fixed time base. The slope of 2.795 means that the stage moved 2.795 μm per sample point. Notice there is a slight third order component to the spot position and a slight pause in motion toward the end of the travel. However, to a high degree the incident BB moves linearly with the stage motion. This means we can measure the spot position at the extremes of travel as being representative of measurement anywhere in the 400 to – 400 μm region.
Also, the vertical column is not perfectly straight so there is roughly ± 10 μm random shift in the center of the scan as the vertical height is changed. However, if the total BB spot motion is noted and divided by 2, we get the average measured ray height.
Fig. 2. Near linear BB position vs stage motion
Measurements of ray height were taken every 2 mm from 5 mm above the upper ball pole to – 5 mm below. These data are shown in Fig. 3. We took 3 measurements in the ball and 3 in free space to show the change in ray height was linear with axial height in both regions. We did not take a measurement at the back focus of the ball because the BB becomes an annulus at a focus and does not give a ray height.
Fig. 3 Measured and modelled ray heights
Figure 3 is counter-intuitive in the sense that the negative half of the plot is of the ray height inside the ball and those heights are linear with those in free space as though there were no refraction at the glass air interface. Notice, too, that the ray height at 5 mm into the ball is about 0.47 mm, higher than where the ray entered the ball at 0.40 mm. Table 1 shows the measured versus modelled ray heights.
Table 1. Measured and modelled ray heights
Notice the slope of – 0.0693 is close to the Zemax calculation of – 0.0683 of real ray heights versus distance from the ball pole for a ray incident from infinity at a height of 400 μm.
To assure we were correctly taking the data, we would move the stage to ± 400 μm, record the ray height and then remove the ball to be sure the BB was incident on the ball based on the stage position. The position was correct and the error bars on the ray heights are on the order of ± 10 μm or less as seen in Table 1 and the orange curve of the modelled ray heights in Fig. 3.
LENS DESIGN MODEL
We used Zemax to model the ball lens. The model is shown in Table 2.
The final thickness of 5 mm means the microscope focal plane is 5 mm above the ball. To find the ray height at other distances, this value is changed. This model was used to calculate the model values in Table 1. The ray pathes at + and – 5 mm are shown in Fig. 4.
Fig. 4. Ray pathes from Zemax at – and + 5 mm
The ray pathes in Fig. 4 and the full pupil ray trace values in Table 3 help explain the reason it appears there is no refraction at the ball upper pole.
Table 3 shows there is refraction at the top of the ball and that the slope of the exiting ray almost exactly doubles, not a coincidence but the fact the model is a ball lens. If the exiting ray is traced back into the ball as it is if we ask the ray height at – 5 mm, it starts at its height at the pole of .1277 mm and continues at a slope of -.0683 for 5 mm to end up at a height of .4693, as in Table 3 and close to the meaured value in Table 1.
Finally, if we ask at what z height do the incident and refracted ray intersect we find (0.4-0.1267)/ -0.0693 = 0.2733/-0.0693 = -3.944 mm, very close to the pricipal plane of the ball lens.
CONCLUSIONS
We have shown in the case of a ball lens that a Bessel beam propagates through the lens as a single paraxial ray in a lens design program. The ball lens was chosen to illustrate this because it is impossible to introduce an error in the example due to tilt of the lens.
While this is not a perfect experiment in the sense there is about a 1% descrepency between the predicted and measured ray heights, it is difficult to come to a conclusion other than the Bessel beam acting as a single paraxial ray. In doing any sort of experiment the precision of the results depends on the hardware available and the environment. The centroid of the BB can be located to ± 0.2 μm using a 10x objective, a 1 Mp camera and simple centroiding software. The main contributer the descrepency between experiment and theory is related to the stages that position the source, ball lens and microscope.
Clearly there is more work ahead in characterizing other examples of lenses and glass indices. While not reported explicitly in this paper, we have done experiments with a ball of a different index and at other axial distances. None of these results are inconsistent with the conclusion here.
In addition to reporting on other lens shapes including aspheres, and other indices, a theoretical verification of physical optics ray propagation is needed.
We end by a few words on the potential future impact of these results. For years lens systems have been designed based on multiple instances of tracing single rays but in the end the only method of accessing the alignment of the end product is based on the collection of all the rays filling the lens aperture. It was never possible to interrogate a single ray to see if it were going where it was predicted to go.
By being able to define position and angle of a ray incident on an optical element or system and predicting where the ray will exit in position and angle, as we can from lens design or ABCD matrix algebra, we can locate the optical system in 5 degrees of freedom in space to a precision consistent with our knowlwdge of the ray positions and angles. In addition, the method may allow the wavefront measurement of aspheres and freeforms in a way that incorporates not only the manufacturing errors but the alignment.
ACKNOWLEDGMENTS
We would like to acknowledge the help of two J. C. Wyant College of Optical Sciences students, Karlene Karrfalt and Tyler Collins. One of us (REP) would like to acknowledge the help John Tesar, an independent optical consultant in Tucson, and anonymous funding sources that have forced him to think harder about alignment.
REFERENCES
Parks, Robert E. “Rapid centering of optics.” Optifab 2021. Vol. 11889. SPIE,
[1] Durnin, J. J. J. A. “Exact solutions fornondiffracting beams. I. The scalar
theory.” JOSA A 4.4 (1987): 651-654.
[2] Parks, Robert E. “Practical considerations forusing grating produced Bessel beams for alignment purposes.” Optomechanics andOptical Alignment. Vol. 11816. SPIE, 2021.
[3] Nelson, W., et al. “Propagation of Bessel and Airy beams through atmospheric turbulence.” JOSA A 31.3 (2014): 603-609.
[4] Turunen, Jari, Antti Vasara, and Ari T. Friberg. “Holographic generation of diffraction-free beams.” Appliedoptics 27.19 (1988): 3959-3962.
[5] Dong, Meimei, and Jixiong Pu. “On-axis irradiance distribution of axicons illuminated by spherical wave.” Optics & LaserTechnology 39.6 (2007): 1258-1261.
[6] Santarsiero, M., “Propagation of generalized Bessel-Gauss beams through ABCD optical systems”, Opt. Commun. 132 (1996), 1. 2021.
Robert E. Parks and William P. Kuhn Optical Perspectives Group, LLC, 9181 E. Ocotillo Drive, Tucson, AZ 85749
ABSTRACT
We give an example of a Point Source Microscope (PSM) and describe its uses as an aid in the alignment of optical systems including the referencing of optical to mechanical datums. The PSM is a small package (about 100x150x30 mm), including a point source of light, beam splitter, microscope objective and digital CCD camera to detect the reflected light spot. A software package in conjunction with a computer video display locates the return image in three degrees of freedom relative to an electronic spatial reference point. The PSM also includes a Köhler illumination source so it may be used as a portable microscope for ordinary imaging and the microscope can be zoomed under computer control. For added convenience, the laser diode point source can be made quite bright to facilitate initial alignment under typical laboratory lighting conditions. The PSM is particularly useful in aligning optical systems that do not have circular symmetry or are distributed in space such as off-axis systems. The PSM is also useful for referencing the centers of curvatures of optical surfaces to mechanical datums of the structure in which the optics are mounted. By removing the microscope objective the PSM can be used as an electronic autocollimator because of the infinite conjugate optical design.
In the last decade or two the optics community has seen huge strides made in the improvement of optical image quality due to the widespread availability of phase-measuring quantitative-interferometry. Surface topography data from phase measuring interferometers is now commonly used by fine figuring processes such as ion milling1, MRF2 and other computer controlled polishing methods to produce optical surfaces accurate to a few nanometers peak-to-valley. A few decades ago one would have asked “Why do you want surface figure this good?” With the luxury of hindsight we see that some of the applications for highly precise figure include the optics that corrected the error in the Hubble Space Telescope and the ever increasing demands of the semiconductor industry.
With an eye on the past it is clear that if another significant improvement in overall optical quality could be made in optical systems there would be applications waiting for those improvements. However, it is probably unrealistic to assume that the optical figure quality of surfaces can be made much better, or at least better at an affordable cost. On the other hand there is an area where significant improvements can be made; the alignment of the surfaces within an optical system to one another. The same sorts of optical performance improvement that have been made in figure can be achieved by the alignment of optical components to tighter tolerances. What is needed to accomplish this are the tools, and a new way of thinking about achieving better alignment.
There are at least three reasons to think that improvements could be made in alignment. The majority of optical systems are getting smaller which means the absolute tolerances are getting tighter. As the optical tolerances get tighter, the tolerances on mating features of cell and lens get tighter and become prohibitively expensive to manufacture. Finally, using the periphery and seat of an optical element to control centering is operating at the optically insensitive end of the optical lever arm. Optics should be centered based on aligning their centers of curvature directly, again for at least two reasons. The edges and seats of lenses and cells have a poor finish relative to the optical surfaces and it is difficult to impossible to control a tolerance to better than the finish of the part. An optical surface is fabricated well enough to produce a return spot a few microns in diameter at its center of curvature. These spots can be located to a small fraction of their diameter in space and provide the information to align centers of curvature coaxially, or in three dimensional space, to a micron or so.
In this paper we will describe the Point Source Microscope (PSM) 3, an instrument for locating the centers of curvature of optical surfaces to micron accuracy for alignment of optical elements that is analogous to the use of a phase measuring interferometer to provide information used to guide the figuring of optical components. Once the location of the center of curvature of an optical surface is known it is easy to position that center on the optical axis of the system in analogy to what ion milling or MRF can do for surface figure.
The PSM is a video metallographic, or reflected illumination, microscope with a Köhler light source to provide uniform illumination over the field of view. In addition, the PSM has a point source of illumination produced by the end of a fiber pigtailed laser diode that is conjugate to the microscope object surface as shown in Fig. 1 below.
Fig. 1 Light paths in the PSM showing the two sources of illumination and the cat’s eye reflection produced by the point source
Fig. 2 LED array in a 5x microphotograph using Köhler illumination only (left), point source only off a rough surface causing distortion of the retro-reflected spot and a software generated reference crosshair (middle), and both sources of illumination on the printing of a business card showing the retro-reflection from a non-specular surface (right). All images made with Nikon objectives5.
Both sources can be used simultaneously as shown in the right-hand image in Fig. 2. Because the point source produces a retro-reflection, its centroid will always appear in the same pixel location on the video screen but its size (and shape, if the surface is rough) will vary depending on how well the microscope is focused on the surface. A crosshair (Fig. 2, middle) can be aligned to the retro-reflected spot so that if the point source is turned off the location in the image plane where it would appear is known. The PSM could be used with an external fiber source to illuminate a particular pixel location on a surface with an alternative wavelength of light if this were useful. The point source is also useful when trying to image a transparent surface with virtually no defects on which to focus. When the surface is in focus there will be a bright return from the point source retro-reflection even though no other surface detail may be visible in the image.
We have described how the PSM produces a retro-reflected spot when focused on a surface. When the PSM objective focus is at the center of curvature of a concave sphere, light will be reflected from the sphere at normal incidence and produce a focused spot at the PSM objective focus. The same is true for a convex sphere whose radius of curvature is limited by the working distance of the objective. The PSM then relays this spot back to the CCD detector as shown in Fig. 4. The difference between this point image and the retro-reflected spot is that the spot image from the center of curvature is sensitive to the lateral alignment of the PSM to the center of curvature as well as to focus.
Fig. 4 PSM objective focus at the exact center of curvature of a concave sphere (left) and displaced laterally and in focus (right)
The PSM can equally well be used with convex spheres, the only requirement is that the radius of the sphere is less than the working distance of the objective, or that an auxiliary lens is used to create a long working distance as will be illustrated in the example of the doublet below. Because the PSM can be used with convex spheres and cylinders many kinds of mechanical tooling hardware become practical and useful optical alignment tooling. Some examples of this tooling are shown in Fig. 5. Surprisingly, these mechanical spheres and cylinders are very accurate figure-wise and are inexpensive compared to most optical hardware. CERBECTM silicon nitride balls8 are rounder and have better finish than the best chrome steel balls plus are opaque and approximately match the reflectivity of bare glass.
Fig. 5 Mechanical tooling hardware including cylinders (plug gauges and locating pins), spheres (bearing balls and tooling balls), and plane mirrors (gauge block target mirrors).
Finally, it should be noted that the PSM also works as an electronic autocollimator when the objective is removed and that is why we have shown the gauge block target mirror among the tooling in Fig. 5. A collimated 6.5 mm diameter Gaussian beam exits the PSM with no objective, is reflected by a plane specular surface and is focused on the CCD detector by the internal tube lens. In the autocollimator mode the angular sensitivity is better than 5 seconds.
3.2 Alignment of a simple doublet lens
This example is given to show how the PSM can be used for alignment in cementing a simple doublet. The optical parameters of this example are such that adequate performance does not require precision alignment; however it is a convenient example to illustrate some of the principles of the PSM. The technique also shows the power of using rotary tables for centering systems with rotational symmetry.
Mindful of the background in Sec. 3.1 on using the PSM at the center of curvature, consider cementing an f/5 doublet objective. Assume the flint is sitting on a cup on a precision rotary table, the surface to be cemented facing up as shown in Fig. 6 (left). This element has been centered with the PSM so that the reflected images from both surfaces are stationary as the table is rotated. The rear (flatter) surface is viewed through the upper surface via an auxiliary lens to converge the light enough to get convergence of the reflected light, in other words, to give the PSM a long working distance to get at the apparent center of curvature. A lens design program is used to find the correct conjugates and, in general, there will be spherical aberration in the return image. If the spherical aberration is objectionably large, the aperture of the lens can be stopped down to the limit where diffraction begins to make the spot larger rather than smaller. The upper surface can be viewed directly at its center of curvature.
A procedure to accomplish this centering is to move the cup laterally until the reflection from the rear surface is stationary. A 1 μm decenter of the lower surface will produce a 2 μm decenter of the reflected image for a total motion of 4 μm when the table is rotated. When the reflection from the lower surface is stationary, slide the lens on the cup until the reflected image from the directly accessible upper surface is stationary as the table is rotated. This procedure should be repeated to be sure that centering the upper surface has not affected the centering of the lower. It is easy to see that the flint element can be centered to better than 1 μm given the PSM sensitivity of 1 μm although there is no need for such accuracy with a relatively slow doublet used in the visible.
Fig. 6 Centering the flint half of a doublet using the PSM and a precision rotary table (left) and centering the crown to the flint (right)
With the flint centered, a drop of cement is placed in the concavity and the mating crown element set in place. Once the cement has been reduced to an appropriate thickness, the crown element may be centered by either of two methods, see Fig. 6 (right). The auxiliary lens may be used to view the center of curvature of the convex surface directly or the PSM can view the reflection off the rear of the flint as seen through the crown. Both methods have similar sensitivity using the parameters of this example but looking directly at the center of curvature is most sensitive. In either case, a 0.01o tilt of the front surface produces a 15 μm or more decenter of the spot that is doubled by rotating the table. If one were to use a contact indicator at the edge of the upper surface, this same tilt would register a 5 μm total indicated runout.
As was explained at the beginning of this section, this example illustrates how to use the PSM for cementing a doublet even though the optical parameters do not warrant this degree of precision and accuracy. There are cases not substantially different from this example for wide field of view projection systems or very fast imaging lenses used in the visible where these sort of centering tolerances are necessary to obtain the desired lens performance. The next example is one where the alignment accuracy is definitely needed.
3.3 Alignment of an Offner relay mirror system
In an example where the need for micron alignment is truly required, consider the Offner9 relay shown below in Fig 8a. Because this is an all reflective system it can be used at a 13.5 nm wavelength in the soft X-ray region and now precision alignment becomes a necessity. The question is how to assemble the two spherical mirrors as well as possible to the mechanical hardware that position the relay optics relative to the rest of the lithography system.
In the example Offner relay used, the total field is about 90 μm in width, the distance from the object plane to the primary is 250 mm and the object is 40 mm off the axis of symmetry. Given this design, Fig. 7 shows the effect decentering or despacing have on performance. Not unexpectedly, performance at the edge of the field is worse than the center but also the image plane is tilted. Knowing the tilt would allow compensating or correcting for the error. Clearly the biggest loss in performance is despace but this can also be corrected by an axial shift in the image plane.
Fig. 7 The effect of decenter or despace on the Strehl ratio of the Offner 1:1 relay in the example
The PSM is particularly useful if some thought to alignment has been made in the initial system opto-mechanical design by incorporating features such as tooling balls to locate critical datums. Assuming critical datums have been defined mechanically, we start the alignment by placing a ball where the two mirrors have their common centers of curvature.
Next, using an auxiliary lens with a focal length slightly longer than the radius of curvature of the secondary mirror, align the PSM focus conjugate with the center of the ball in three directions so that the return image is centered and in focus in the PSM image. Then insert the secondary mirror and align it to the PSM image using the reflection from the secondary convex surface that is conjugate to its center of curvature as in Fig. 8b.
Move the PSM so it faces the primary mirror and align the PSM focus to the center of the ball as in Fig. 8c. Remove the ball and align the primary to the PSM focus so light reflected from the primary is centered on the PSM image and is in focus. Now the two mirrors are concentric and located precisely to the optical bench datum indicated by the ball.
Once the Offner relay is aligned optically and located precisely to its mechanical structure via the tooling ball, light from an object on one side of the system axis of symmetry will be nearly perfectly imaged on the other side of the axis. The relay will have been optimized for a particular object distance off the axis but this distance is not highly critical to the relay performance. It is important to know exactly where the input is imaged on the other side of the axis. If a tooling ball is placed so that its center is at the center of the object field, the PSM can precisely determine where the image will be located by using the relay in double pass, see Fig. 8d. Light from the point source in the PSM is brought to focus by the PSM objective and sent on through the relay until it reflects off the center of the ball located with its center at the object plane of the relay. The light then retraces itself exactly back to the PSM focus where it is imaged by the CCD and its location shown on the video monitor. When the return spot is centered in the reference cross hair and is at best focus, the PSM focus is in focus at the center of the relay image plane. This check of object and image location also permits a double check that the alignment was done correctly, something that is always useful when the tolerances are tight.
Fig. 8a Offner 1:1 relay wit ball at center of curvature
Fig. 8b Secondary aligned to PSM via auxiliar1y lens
Fig. 8c Primary aligned directly to ball
Fig. 8d PSM locating Offner center of image field
We conclude this example by saying the Offner 1:1 relay is the simplest sort of all reflecting, ring field microlithography optical systems. The all reflective systems that will be used to make future IC chips will contain six or more mirrors with their centers of curvature located along a common optical axis.10 If these systems are to meet their theoretical optical design performance, they will have to be assembled so their centers of curvature truly lie on the optical axis to 1 μm or better tolerances. Variations on the method described here will make this possible and avoid the inevitable tolerance stack up when optics are centered mechanically by their edges and seats.
3.4 Alignment of an off-axis telescope system
In this example we will describe two other modes of using the PSM for optical alignment and alignment of optics to mechanical datums. In practice, though not often thought of this way, all optical alignment deals with positioning optical surfaces in a particular relationship to mechanical datums as in the case of centering a single lens element. The optical axis is the line joining the centers of curvature of the two optical surfaces. Edging a lens is bringing its periphery, a mechanical datum, into concentricity with the optical axis. In this example, the telescope system is similar to the single element with the telescope itself being like one optical surface, the backend of the system like the second surface and the mounting plate for the system like the periphery of the lens. In this case, the periphery was established first and the two halves of the optical system each brought into alignment with the periphery or mounting plate. Figure 9 shows a simplified outline of the telescope system.
Fig. 9 Simplified view of the off-axis telescope system in the example
The telescope on the left of the mounting plate is an off-axis Ritchey-Chretien with an f/11 output as shown in Figure 9. The telescope and the prism housing are attached to the mounting plate by a three ball and groove kinematic mounting scheme, one coupling of which is shown in Fig. 10. Drilled balls fit into conical countersunk recesses in the telescope frame and prism housing. The two halves are pulled together by a screw through the center of the balls and are located by a pair of pins forming a double sided “V” groove in the mounting plate. In this way the two halves of the telescope are precisely located relative to each other and flexing of the mounting plate (by which the telescope assembly is held in its housing) will not disturb the alignment of the telescope system. Test fixtures for both sides of the telescope system used the same mounting scheme to accurately locate the halves during individual assembly and test.
Fig. 10 Kinematic mount detail showing one mounting point with two pins forming a double sided “V” groove
3.4.1 Alignment of the telescope
By design the optical axis of the telescope was precisely defined relative to the telescope frame. The telescope frame, in turn, was precisely located relative to the optical table on which the telescope was aligned by means of a fixture incorporating the three ball kinematic mounting scheme as shown in Fig. 11. Two ends of the temporary long screws can be seen in the upper kinematic locators clamping the telescope frame to the test fixture.
Fig. 11 Telescope frame held to the test fixture via kinematic mounts. The off-axis primary mirror is visible at the left end of the frame
Prior to aligning the telescope, the telescope alignment test fixture was adjusted in four degrees of freedom relative to the collimator; two degrees of tilt to make it square to the collimated beam and two degrees in translation to center the apertures. Then two balls were placed on the table relative to the test fixture and table so as to provide reference positions accessible to the PSM at the short focus of the hyperbolic primary and at the system focus. The line joining the centers of these balls defined the optical axis of the telescope mechanically
The telescope mirrors were then aligned in two steps against a collimator, first the primary alone and then the combination of primary and secondary. Initially the PSM was positioned so that it was looking into the center of the primary with its focus at the center of the ball at the primary focus. The collimator was then de-focused to produce a focus far behind the primary at the long focus of the hyperbolic primary. The source in the collimator also had to be raised precisely because the optical axis of the collimator and telescope were at known but different heights.
Once the PSM was aligned with the ball defining the location of the primary short focus, the ball was removed so the PSM could view the light from the collimator and focused by the primary. It was necessary to adjust the primary in five degrees of freedom, three to correctly position the focus and two to remove the aberrations caused by the primary optical axis not lying on the collimator axis. First the primary was adjusted so that light from the collimator and reflected by the primary was in focus and centered on the PSM crosshairs and in the computer monitor. This process aligned the first three degrees of freedom but the image showed substantial astigmatism, exactly what would be expected from a misaligned off-axis hyperbola.11 (It should be noted that since this was an off-axis telescope the aberration was largely astigmatism with a little coma and the two aberrations decreased together as the primary was aligned. Had this been a symmetrical telescope the only aberration seen due to misalignment would have been coma.)
The next step was to adjust the primary tilt and decenter to maintain the image centration and focus on the PSM crosshairs and to reduce the astigmatism. The most straight forward method of doing this is tilt and decenter sideways to straighten the astigmatic image so it is perpendicular to the optical table. Then adjust vertically relative to the table to reduce the length of the image until it is round and symmetrical. This procedure worked in a quick, directed fashion. The images observed with the PSM corresponded to the images in reference 11. Further, the PSM had sufficient magnification to view the “star” image to reduce the astigmatism to at least the tenth wave level. The star test is very sensitive to symmetry in the image and by going through focus the least astigmatism is very apparent.
Once the primary was adjusted so that it focused in the correct location and was free of aberrations indicating that its optical axis and the mechanical optical axis were coincident and parallel to the collimator axis, the PSM was mounted looking into the telescope centered on the secondary mirror aperture with its focus centered on the ball defining the system focus. Alignment of the secondary followed the same sequence as the primary but now the collimator was set at infinity focus with its source on axis. Fig. 12 shows this phase of the alignment with the PSM barely visible behind the telescope while the monitor shows the image from the telescope in the crosshair. The kinematic base at which the ball screwdriver is pointing accepts the posts (lying on their sides) that hold the balls that define the locations of the primary and secondary foci.
First the secondary mirror was adjusted in tilt and decenter so that the image was centered on the PSM crosshairs and was focused as well as could be determined given the residual astigmatism. Then the secondary was adjusted using a combination of tilt and decenter to reduce the astigmatism to give a well focused, symmetrical image while keeping the image centered on the PSM display crosshairs. Once this alignment had been performed we knew that the telescope focus was located the correct distance behind the test fixture and that the optical axis of the telescope as defined by the line between the primary focus and the telescope of secondary focus was perpendicular to the test fixture and correctly positioned relative to the three kinematic mounting locations on the test fixture.
The main difference between this telescope alignment technique and those described previously is that by using aberrations as well as locations, the PSM can monitor alignment in five degrees of freedom, something that would be impossible using an alignment telescope. Further, because the PSM operates with a relatively fast cone of light it is much less confusing than an alignment telescope in terms of selecting the reflection from the surface of interest. The alignment of telescope took less than a day and the secondary, which had more refined adjustments on the mount, took less than 1⁄2 hour including PSM setup time.
3.4.2 Alignment of the prisms in the prism housing
There were numerous prisms and fold mirrors in the prism housing to direct light from the telescope to various sensors as well as a path to direct light into the telescope and a path to couple a display into the eyepiece. All of these paths had to be boresighted so that as the operator changed from one mode of operation to another an object in the center of the field for one sensor remained centered for another. Further, it was desired that the prism housing and sensors be interchangeable with the telescopes so the center of the field had to be defined relative to a mechanical datum, namely the mounting plate coupling the telescope to the prism housing. To align these various paths two fixtures were used, a projector that produced a converging cone of light that matched the telescopes f/11 focus and a mounting fixture for the prism housing that exactly simulated the mounting plate. This approach is shown schematically in Fig. 13 where the ball in the foreground represents the projector source; the joined base of the cones represents a positive lens to produce the f/11 cone that ends with the far ball at the conjugate focus.
Fig. 12 Secondary mirror alignment phase with PSM behind the telescope, the image spot on the PSM computer monitor and kinematic locating bases for the posts (lying on their sides) and balls that define conjugate locations
Fig. 13 Location of the source projector relative to the prism housing (left image)
Fig. 14 The various optical paths produced by the prisms in the housing with a ball centered at the focus of each (right image)
The three balls in the prism housing mate with the test fixture to hold the prism housing correctly in five degrees of freedom. Three degrees of freedom are controlled by the ball at the projector focus. The other two degrees of freedom control the location of the optical axis of the projector relative to the three balls defining the prism housing location. This is exactly the same condition that applied to the telescope so that when the two halves of the system are mated the telescope will focus at the far ball location and the telescope optical axis will be co-axial with the projector axis.
Figure 14 shows the prisms in the housing and the paths into which the prisms direct the beams. These beams need to be correctly positioned to a few pixels on each detector so the final alignment can be accomplished by a very slight lateral adjustment of the sensors themselves. To accomplish this, the test fixture that held the prism housing was drilled and bushings inserted directly beneath the foci of the various paths and rods of the correct lengths were cut to hold balls at the foci as shown in Fig. 15, a picture of the actual fixture set up for locating one of the paths.
Fig. 15 Prism housing mounted to the test fixture with ball on metering rod to define the path focus and the PSM adjusted using the x-y-z stage so the its objective focus is at the center of the ball in the three translational degrees of freedom
Figure 15 is instructive in that it shows almost all of the elements of the alignment discussed to this point. The f/11 light cone is coming in from the left and going through the prism visible in the prism housing that is attached to the test fixture via the three kinematic locators clearly visible on the back of the fixture. A metering rod is located in one of the bushings and has a ball sitting on its top. The PSM is aligned pointing down the diverted beam path and the PSM objective focus is aligned to the center of the ball. The PSM is supported by a substantial x-y-z stage to facilitate its alignment with the ball center. The PSM tilt adjustments are relaxed because they only control vignetting.
The next step in the alignment of this beam path was to remove the ball and metering rod so light diverted by the prism could be viewed by the PSM. Crude adjustment of the prism was made with the projector source set at maximum intensity. The focus of the projected beam was easily visible on a card in front of the objective and the prism was tapped lightly to tilt the prism and bring the beam focus in front of the objective. Once the beam entered the objective the source intensity was reduced and the beam position monitored on the computer display. The real-time visual feedback of spot location and focus made positioning the prism straightforward and rapid. Another feature of the PSM allows left/right and up/down image flips so the displayed image always had the expected or natural orientation independent of mirror flips of the beam. This ergonomic feature make alignment easier because adjustments appear to work in the direction expected instead of the way they actually move.
Once the prism had been adjusted so the spot of light was positioned within tolerance, UV cement was carefully applied to several places along the edges of the prism and the prism tacked in place with a UV source. Once tacked and still properly aligned, epoxy cement was applied and allowed to cure slowly. The next prism in the chain was added, the proper metering rod located in the next bushed hole and the procedure repeated for the remaining prisms in the paths. The procedure for alignment of the six paths in the prism housing was so efficient that all the paths could be aligned in two hours. In addition to the time savings, the mechanical tolerances on the prisms and their holders did not have to be very tight. Ledges to guide the initial prism placement were all that was required. The balance of the alignment was done by slight taps and shimming to correctly position the prisms.
3.4.3 Alignment using the PSM mounted on a CMM
It should be noted that there is an analogous approach to aligning the prisms that has been very successfully used OPG on another off-axis optical system, that is to mount the PSM on the ram of a coordinate measuring machine (CMM) and use the CMM to position the PSM at the end of each path rather than use the fixture as we have just described12, 13. Generally the CMM approach is more effective when there are either only a few prototype systems or a variety of systems to align while the fixture method is more suited to production. The geometry and complexity of the system may influence the decision as to which approach would be best as well as the availability of a CMM.
Using the CMM eliminates two aspects of the fixture. Using the example of the prisms above, a source projector is still required but the location of the source focus and the position of the optical axis can be determined using the PSM and stored in the CMM. To accomplish this, the first step is to mount the PSM on the CMM ram in place of a touch tip probe and set the probe tip radius set to zero in the CMM software. Next the PSM objective is focused at the center of the CMM master ball and the coordinate system zeroed in all three directions simultaneously. Then the PSM is moved to the projector source and the PSM focus made coincident with the source focus. The coordinates of the source are noted. Then the PSM is moved to the image of the source and that location noted. This defines the five degrees of freedom of the source location as shown in Fig. 16, four degrees define the axis and one is the axial location of the focus.
Fig. 16 PSM used in five positions on a CMM to locate the source projector axis and focus (2 – left facing), and prism housing (3 – right facing)
Now the PSM would be rotated 180 degrees about a vertical axis and the PSM focus zeroed again on the master ball. Then the locations of the three balls defining the location of the prism holder can be found, see Fig. 16. The telescope mechanical design defines where these three balls must be relative to the optical axis and focus of the source projector to be aligned. Using COTS alignment stages the prism housing can be brought into alignment with those three positions to locate it correctly relative to the source projector. Once the housing is correctly positioned, the PSM can be rotated to look down the first path to be aligned, zeroed again on the master ball and moved to the first path focus ready to sense the position of the first beam. The CMM software displays the global coordinates of PSM objective focus after each zeroing on the master ball. All that is necessary is to read from the design where the next focus is and move the PSM to that location and lock the CMM axis motions until the prism is aligned. Once the concept of using a CMM as an assembly tool is understood, the implementation is straight forward, and the CMM takes on added value as an assembly tool in addition to its traditional use as an inspection tool through use of the PSM.
Another way of aligning an optical system with powered elements on a CMM is to place the optical bench onto which the elements will be mounted on the CMM bed. Assuming provisions have been made in the opto-mechanical design, after zeroing the PSM on the CMM master ball, pick up three known tooling ball locations on the optical bench. Since these locations will be known by design relative to the optical axis and centers of curvature of the various optical elements, the PSM objective focus can be located precisely as each element is added. By picking up the three tooling balls on the optical bench, the CMM software establishes a coordinate system in the optical bench space and displays x- y-z coordinates in this space. When the first optical element is introduced the PSM can be located at the centers of curvature (or apparent centers of curvature) of the two surfaces. The element is then adjusted by tapping or shimming until it is correctly located within the given tolerances. Again, neither the element nor the holder need to be machined to tight tolerances because the element location can be adjusted far better using its optical axis, which is the line joining its centers of curvature. This alignment method largely eliminates tolerance stack up because the usual glass-metal (or optical-mechanical) interface features are not used to define optical alignment.
4. COMPARISON OF THE PSM WITH OTHER OPTICAL ALIGNMENT TOOLING
Now that various methods of using the PSM for the alignment of complex optical systems has been explained, we compare the features of the PSM with other more familiar optical alignment tooling. First, it should be said that the PSM is a compliment to existing tooling, not a replacement for it. For example, in Figs. 12 and 15 an alignment telescope is clearly visible. At the same time, there are other situations where it is very handy to have more than one PSM available.
4.1 Autocollimators
Although no examples have been given of the PSM used as an autocollimator, it is one if the objective is removed because the microscope is based on infinite conjugate optics. With the 100 mm tube lens and tenth pixel sensitivity, the PSM has angular sensitivity of about 5 seconds in reflection over a 6 mm beam diameter. While this is almost an order of magnitude worse than what can be achieved with a Davidson14 or Nikon15 autocollimator, there are situations where this electronic autocollimating capability is very useful, particularly in cramped locations and for use in conjunction with a CMM to move the PSM without pitch or yaw from interrogating one plane surface to another for parallelism.
An ergonomic feature of the PSM in this mode is the display on a computer monitor as opposed to looking through an eyepiece that might not be conveniently located. A very practical advantage of the PSM used as an autocollimator is that it is easy to align initially under normal lab lighting because of the bright laser source. With the more traditional autocollimators it is often difficult to find the return beam initially even if the lab lights are turned off.
4.2 Alignment telescopes
The PSM is probably most analogous to an alignment telescope in terms of what it does best but direct comparisons are difficult. A typical alignment telescope locates a line of sight to about 1 part in 800 of the full field at the object distance that the telescope is focused. This works out to roughly 5 seconds, the same as the PSM used as an autocollimator. The PSM has a minimum of 760 pixels across the field and is sensitive to 0.1 pixels when centered on a spherical surface and is thus about an order of magnitude more sensitive than the alignment telescope. The alignment telescope can bring objects on that line of sight into focus one after another as the focus is changed from near (about 16”) to infinity. This is very convenient when all the objects are in a line because no part of the test setup needs to be moved during the alignment. On the other hand, the alignment telescope is nearly useless if the objects needing alignment are in a three dimensional space such as the paths in the prism housing example or in aligning the components of a spectrometer.
The PSM is perhaps best thought of as a null or end point detector in three dimensions and it does this to about 1 μm laterally and a few μm axially. The ideal way to use the PSM is either mounted on the ram of a CMM where it can locate spherical and cylindrical features to better than the accuracy of most CMM’s over any reasonable distance, or in conjunction with a fixture like that described for aligning the prisms. In both cases, the PSM can locate centers of curvature and axes of cylinders to as well as or better than the mechanics of the CMM, or fixtures can establish the desired mechanical datums in three dimensional space. In this sense the PSM has a sensitivity of at least an order of magnitude better than an alignment telescope. Coupled with its small size, light weight, ability to be used as an autocollimator and ergonomically convenient to use display, the PSM can turn tedious assembly and alignment operations into rapid, accurate and easily documented procedures.
5. CONCLUSIONS
We have described the Point Source Microscope and illustrated its usefulness in the alignment of various optical systems by accurately linking mechanical datums to optically functional surfaces and these surfaces to each other. In comparison with other optical tooling, the PSM is a flexible compliment to, rather than replacement for, traditional autocollimators and alignment telescopes. A combination of the PSM’s accuracy and its use aligning the optically significant features of an opto-mechanical system make it possible to achieve higher alignment accuracy than attempting alignment via strictly mechanical features. This leads to systems with better optical performance than can presently be achieved at a reasonable cost.
With insight as to how the PSM works it is clear that the PSM, in a very practical sense, makes many alignment operations easier and quicker to perform to a higher level of accuracy, particularly for optical systems that are distributed in three, rather than two, dimensions, or do not have circular symmetry. Although difficult to document specifically, we have seen the time taken to align complex optical systems drop by a factor of ten while achieving better alignment than more traditional methods. Furthermore, by aligning using the optically functional surfaces of a system, both tolerance build up and the expense of precisely made parts can be largely eliminated.
6. ACKNOWLEDGEMENTS
The telescope referred to in Sec. 3.4 was a team effort of Breault Research Organization, Inc (BRO)16 and Optical Perspectives Group, LLC (OPG). The lead optical engineer on the project was Matthew Dubin of BRO who had many insightful and useful comments regarding the use of the PSM. OPG’s primary responsibility was assembly and test of the system that was performed at BRO with the able assistance of Joe Barcelo. The optical design of the telescope and sensor optics was performed by Richard Buchroeder of Optical Design Service under contract to BRO. The mechanical design of the telescope was done by Kris Tvedt of BRO. The details of the kinematic location features were due to Bryan Loucks, an opto-mechanical design consultant working with OPG.
10 D. M. Williamson, “Evolution of ring field systems in microlithography”, Proc. SPIE, 3482, 369-76, (1998).
11 R. E. Parks, “Alignment of off-axis conic mirrors”, Optical Fabrication and Testing Workshop Technical Notebook, OSA, Falmouth, MA Sept. 1980, pp. 139-45. Revised reprint available at www.optiper.com.
12 R. E. Parks and W. P. Kuhn, “Using a CMM for optical system assembly and alignment”, ASPE Proceedings Spring Topical Meeting on Coordinate Measuring Machines, 29, 3-8, (2003)
From the beginning we have said that centering a lens meant aligning the optical axis of the lens to a reference axis. Up to this point we have assumed the axis was the mechanical axis of a rotary table. Because there are disadvantages to using a rotary table, I was curious to see if there was another way to create a reference axis. The path to a new axis came from the subject we have been discussing, the cementing of doublet lenses. The idea also came from my interest in computer generated holograms (CGH) for use as calibration artifacts for coordinate measuring machines and CNC machine tools. [1]
As we have discussed, the seat of the convex side of the meniscus half of a doublet determines the location of the center of curvature of that side. The center of curvature of the concave side is the second point defining the lens axis. It is easy to design a CGH of Fresnel zone patterns to serve as datums for cementing 3 balls to the CGH centered on a Fresnel zone of the correct radius to match the location of the center of curvature of the concave side looking through the lens. [2] A combination of the seat and center of curvature make precision centering the meniscus easy when a centering sensor is centered on the center of curvature.
A second Fresnel zone of a different radius is designed to focus at the same height when the positive element is placed on the meniscus. This means that the meniscus is first aligned and fixed in place and then the positive element aligned to the meniscus without ever having to move the alignment sensor. If the sensor does not have to move it cannot introduce errors due to its movement. Top and side views of such a CGH and doublet alignment scheme are shown in Fig. 1.
As you can read about the successful experiment in the reference there is no need for the details here. While the CGH did exactly what was intended it meant that a different CGH would be required for every different doublet design. Also, this design using balls to serve as a seat made it difficult to secure the aligned meniscus so it would not move when adding and aligning the positive element.
This success coupled with the use of a zone plate CGH to measure the straightness of travel of mechanical ways on machine tools [1] lead to the idea of using the “beam” of light created by the zone plate as a universal method of centering doublets and other alignment tasks. The concept of using a zone plate for alignment purposes is not new. In the 1950’s the Dutch optical scientist A. C. S. van Heel discussed the problem of aligning multiple points to a line because it is difficult to see all the points at once [3]. He goes on to suggest using diffraction by a slit for alignment in one axis before generalizing the idea to use a zone plate of evenly spaced concentric circles. He says that under good observing conditions alignment to 1 µradian is possible. [4]
In 1954, John McLeod of Eastman Kodak, also interested in alignment, invented the Axicon [5] in part to improve the light efficiency of alignment instruments. He said that by illuminating the axicon with a pinhole source of light it “will project a continuous straight line of image out to infinity.” In Fig. 1 of his paper, he shows a diagram that is very close to that in the paper by Durnin [6] that shows how an illuminated annular mask creates a beam of light coined by him as a Bessel beam.
Technology has made great progress since the days of van Heel and McLeod. The zone plates van Heel used were made by light grinding of rings in a polished glass plate to create rings that would transmit and block the light. The rings were typically 1 mm in width. Later the rings may have been made photographically but I could not find a reference. McLeod’s axicons were not the easiest optics to make either, and it wasn’t long after Durnin’s paper that Turunen, et. al. [7] showed that it was easy by then to make zone plates as computer generated holograms (CGH) with narrow line spacings.
It turns out in hindsight that both van Heel and McLeod were producing and using Bessel beams for alignment purposes. It took about 30 years for theorists to review their experimental results and add a theoretical explanation to the phenomenon described by the earlier empirical observations.
This is a good place to describe a simple Bessel beam as it turns out there are many variations. The simplest beam has an intensity distribution perpendicular to the direction of propagation proportional to the square of the 0 order Bessel function as in Fig. 2 (right), and the intensity distribution captured on a digital camera in Fig. 2 (left). The first ring of the pattern has 16% the intensity of the central peak, and the peak and each ring has the same radiant energy. This means it is easy to precisely centroid on the peak with a digital camera to a small fractional of a pixel if the peak covers a diameter of 3–4 pixels. Because the irradiance of the pattern is nearly constant, a quadcell is useless for centroiding.
Most discussions of Bessel beams assume the illumination of the axicon, or zone plate is collimated. With collimated illumination the beam is a finite length. A point source of light is also a suitable source and produces a beam of close to infinite extent. The source does not have to be coherent or monochromatic. White light works fine. Each of these changes gives a slightly different formal name to the kind of Bessel beam and the ones I use by illuminating a grating with the end of a single mode fiber and laser diode source should probably be called Bessel-Gauss beams.
Now that I have described what a simple Bessel beam is and how they are produced we should get back to the way they propagate through optical systems and why they are useful for alignment. After some initial experiments over 3 years ago using the Bessel beams to measure the straightness of ways on machine tools and realizing errors in transverse motion of a fraction of a µm were easily observed, I began watching what happened when I moved a lens through the beam. I found that when my ASM was focused inside a lens at what appeared to be the principle plane, the transmitted beam did not move as the lens was moved perpendicular to the beam. It did move one way above the plane and the other on the opposite side of the apparent principle plane. This is paraxial rather than real ray behavior.
In the meantime, I found the paper “Propagation of generalized Bessel-Gauss beams through ABCD optical systems” [9] that implies the behavior I was seeing was paraxial. This prompted me to do an experiment with a ball lens and create the Bessel beam with a 50 lp/mm axicon grating from AOM. [10] A ball was used since it is always free of tilt and all I had to control was the decenter, or ray height, of the Bessel beam incident on the 8 mm diameter, BK7 ball.
In the experiment, [11] the ball lens and microscope were stationary, and the Bessel beam was translated across the ball close to the ball center to keep the experiment paraxial. For this BK7 ball with an index of 1.515 the efl is 5.8834 mm. If the ball is decentered 1 µm the ray will exit the ball at an angle of h/efl = .001/5.8834 = .00017 radians. The intersection of the entering and exiting rays was at the mid-plane of the ball, its principle plane. The experiment confirmed the paraxial behavior of the Bessel beam.
This paraxial behavior has profound consequences relative to alignment and to measuring lens properties in general. The transverse position of the beam can be measured anywhere along the beam including inside the ball, lens or system of lenses. Or far from the lens so a small angular deviation causes a large transverse displacement. Just like the classical textbook picture, Fig. 3, from geometrical optics, if the incident ray is parallel to the lens optical axis it changes direction at the second principle plane to cross the optical axis at the focal point of the lens. (The region right around the focal point is the only place the Bessel beam cannot be viewed because it turns into an annulus.)
Because the Bessel beam behaves as a single paraxial ray it is easy to calculate the ray deflection angle due to decenter as simply the decenter over the efl. This immediately gives an idea whether a particular height above the lens will be sufficient to give the required sensitivity. If the lens sits in a centered seat, then tilt of the lens changes the ray angle by (R1/efl) times the lens tilt.
For our example meniscus a decenter of 10 µm changes the ray angle by .01/-110.66 = -9.04 µradians and a lens tilt of 1 mradian changes the ray angle by (264.97*.001)/-110.66 = 2.394 mradian. Not surprisingly lens elements with short focal lengths are more sensitive to misalignment than ones with longer focal lengths, or less power.
We have covered many topics in this Chapter, so it is time for a quick review.
· The expense and tediousness of using a rotary table for lens centering led me to look for an alternative method
· An initial use of a Fresnel zone CGH was successful centering but not practical
· Experience with zone plates made with CGH techniques for mechanical alignment led me to try them for optical alignment
· The Bessel beams made by the zone plates appeared to propagate through optical elements and systems as though they were paraxial rays
· Published works and further experiments convinced me the propagation was paraxial
· This meant the Bessel beam position could be measured anywhere along the beam
· In turn this means it is easy to measure the beam angle exiting a lens to find centering errors independent of where center of curvature are located
In the next Chapter we will look at more consequences of the paraxial behavior of Bessel beams and their application to centering and alignment.
References
[1] Parks, R., Ziegert, J. and Groover, J., Computer Generated Holograms as 3-Dimensional Calibration Artifacts, Proc. ASPE, 117–20 (2017)
A convenient and concrete example of aligning three centers of curvature is the cementing of the doublet we used in the optical axis example in Chapter 5, the details of which are reproduced below.
Fig. 1 Doublet used as an example of aligning three centers of curvature
The goal is to get all three centers of curvature on the same line to some tolerance dictated by the desired lens system performance. When alignment and centering is done in practice the meniscus element is placed on a seat with the concave surface facing up so that when a drop of cement is added it is contained within the concave well. We have already been over the details of aligning the optical axis, or two centers of curvature, of the meniscus to a seat in Chapter 7.
As seen in Fig. 1, the task now is to align the upper convex surface of the positive element so that its center of curvature lies on the optical axis of the meniscus. The interface between the two lens elements takes care of itself as the cement acts as a lubricant between the interface surfaces. The cement is worked out to a thin uniform thickness by pressing down and gently sliding the positive element around on the meniscus. Once the cement is thin enough that there is a drag resisting the sliding the positive element is centered and the cement cured.
Because the surface facing the alignment sensor and its objective is convex, the objective must have a working distance of more than the radius of the upper surface, or roughly 71 mm in this case. Unless the diameter of the lens being cemented is quite small, the convex surface will be longer than most commercially available microscope objectives. Autocollimator type centering sensors come with a variety of working distance objectives and autostigmatic microscopes can be fitted with simple doublet objectives to get the required working distance.
While using an objective with a long enough efl will almost always reach the center of curvature, there is a loss in sensitivity to alignment due to the increase in the focal length of the objective. For cementing doublets this is probably not an issue because the longer the radius, the weaker the lens and the less sensitive the lens is to alignment. There are cases where the doublet has a large diameter and is relatively fast where alignment can be an issue so there is a need for greater sensitivity. There are other occasions where there is physical interference, and a long working distance is needed to reach the center of curvature. In these cases, it is possible to add a 1 to 1 relay in front of the objective that has sufficient sensitivity. An example of such a relay for use with an autostigmatic microscope is shown in Fig. 2. This makes an aesthetically ungainly instrument but achieves the purpose of no loss of sensitivity.
Fig. 2 Long working distance objective with same lateral sensitivity as the microscope objective
Going back to our example, assuming we use an objective with a 100 mm efl to reach the center of curvature of the upper surface, we have a situation as in Fig. 3. The 0.5° (0.0087 radian) tilt of the positive element moves its center of curvature 1.042 mm off the axis of the meniscus. Since almost any optical sensor can detect a µm decenter and there is a 2x magnification of the displacement due to reflection, the center of curvature location is easily measurable to 1 second of arc, better than any but the most stringent centering requirement.
Fig. 3 Light from a 100 mm efl objective reflecting from the 0.5° tilted positive doublet element showing the displaced center of curvature of the convex surface rotated about the interface surface
Figure 3 shows the mechanical implications of being able to center to 1 second of arc. From similar triangles, the positive element is about 1/3 of the distance from the objective to the center of curvature which implies that the edge of a perfectly centered element will be aligned to a perfect meniscus to about 1/3 of a µm, far better than such an element is realistically edged.
What I am trying to illustrate here is that aside from very small (cell phone camera and endoscope lenses) and very large lenses, relying on mechanical tolerances of the edges of elements may not be the optimum approach where precision centering is needed. Optical centers of curvature are the critical features for alignment, not mechanical interfaces at the edges. The peripheral mechanical interfaces should be designed to allow for the adjustment of the centers of curvature, but these interfaces should not be the features that define the alignment when high precision is needed.
As an example, some doublets are centered and cemented using a “V” block to align the edges of the elements. Assume we want to align our example doublet to 3 minutes of arc or about 1 milliradian, a typical centering tolerance for good but not precision centering. This means that the 2 elements must be edged within 50 µm of the same diameter to meet this tolerance assuming no other errors in the centering process. If there is any wedge in either of the elements there will be additional errors.
I think the idea of the use of optical versus mechanical features for alignment goes beyond the optical elements themselves. Consider the typical rotary table centering device. It uses a mechanical axis of rotation to define the reference axis to which we want to align our optical elements. In one sense this is an ideal method of centering as we have previously illustrated; if the transmitted or reflected image from an optical surface moves it does not lie on the axis of the rotary table, full stop. However, we have optical centering sensors that detect motion of an image to a fraction of a µm. That means the runout of the table must be less than a µm to make use of the full sensitivity of the sensor.
In addition, the tilt or wobble, of the table must be small and the further the optic being centered is above the table the worse the effect may be. Assume our table has a 1 second of arc (5 µradian) tilt and the optic we are aligning is 100 mm above the table. As the table rotates the mechanical vertex of the element will translate ± 0.5 µm and if our sensor could see a mark at the vertex, it would be seen to move 1 µm TIR (total indicator reading, a terminology from mechanical touch indicators).
However, our optical sensor is not viewing the vertex of the element but rather the center of curvature. In the case of our example doublet we have the situation in Fig. 4 where the center of curvature of the upper surface lies near the rotary table surface.
Fig. 4 Misaligned doublet on rotary table having a tilt error
Since the center of curvature lies near the table and we assume that the tilt in the table occurs near the table surface, the motion of the center of curvature is reduced by the ratio of 29/100. The reverse is true of the interface surface, its CoC moves more by the ratio 148.5/100 and the CoC of the lower surface of the meniscus by 356/100.
To minimize the errors in rotary tables they almost always are equipped with adjustment screws to remove the tilt and decenter that rotate in synchronism with the table. Errors of a higher frequency than once per revolution cannot be removed by these adjustments, however. The precision of alignment using a rotary bearing as the reference axis is never better than the quality of the bearing.
If the quality of the rotary bearing limits the precision of centering, what if the lens remained stationary? Then the quality of the vertical slide becomes the limiting factor because you must rely on its precision to move the sensing unit from one center of curvature to the other to establish the axis of the lens. This is not a new idea and was proposed in a paper1 a few years ago. The paper points out the advantage of this method. Assuming a sufficiently high precision vertical stage, the method is much faster than using a rotary table because there is immediate feedback of the result of making a centering adjustment.
When a rotary table is used you make an adjustment and then rotate the table to see if the adjustment has improved the centering. If the lens is stationary, you are making the adjustment to drive the reflected image to a particular position in the sensor coordinate system and can directly see how fast and far the reflected spot is moving without worrying about rotating the table. The paper estimates that eliminating the rotary table would speed up the alignment process by a factor of 5-6. To the best of my knowledge a product has never been brought to market based on this method. This may indicate that the method is not easy to implement in this form despite its apparent advantages.
The idea has merit, however. It depends on how the vertical slide is designed and implemented. Think of the possibility of using the vertical ram (the arm of the machine the probe is attached to) of a coordinate measuring machine (CMM) as the vertical slide. Even relatively inexpensive CMMs read out the probe location to ± 1 µm. This does not mean the probe moves with µm straightness, but that the calibration method and software tell you where the probe is in 3 degrees of freedom (DOF) to on the order of ± 1 µm plus a factor for the distance moved. Compared to the price of a precision centering device with a rotary table, a CMM with an optical sensor may be a cost effective method when it is balanced with productivity improvement. You only purchase the hardware once; you pay for labor every day.
Using a CMM for centering2 still means you must raise and lower the ram to go from one CoC to the other. The adjustment at each CoC is much faster than with the rotary table it but will still take more than one look at each CoC to make sure the best alignment is achieved. Further, the method still depends on precision mechanical motion and calibration. What if centering to remove tilt and decenter could be done without having to move any part of the centering measuring device? What if the only adjustments were the 4 that adjust tilt in 2 DOF and decenter in 2 DOF. The method would be faster and more precise.
I have hinted at the method in Chapter 5 where we define the optical axis of a multi-element system as a line in space representing a single transmitted ray that is unchanged in location or direction when the lens is inserted in the ray path. In the next Chapter I will talk about how to generate a single ray that is useful for aligning optical systems in general but specially for centering of lenses to an optical reference axis.
1 J. Heinisch, F. Hahne, P. Langehanenberg, “Rotation-free centration measurement for fast and flexible inspection of optical lens systems,” Proc. SPIE 11175, 111751B (2019)
In the previous chapter we looked at finding a single center of curvature using any of several optical instruments. This locates a particular point in space but does not define an axis. For that, two centers of curvature separated by a finite axial distance must be located to define an axis, or line. This chapter is devoted to finding an axis based on locating two centers of curvature simultaneously.
We resume with the assumptions from the previous Chapter that we have a rotary bearing table with its axis vertical and our optical sensor centered on the axis of the table looking downward at the table as was shown in Fig. 1 of Chapter 6. The sensor must be mounted on a vertical slide moving parallel with the table axis to reach the two centers of curvature.
(Sidebar – Presumably an alignment telescope could be used to view the centers of curvature by adjusting the focus knob rather than moving the instrument along a vertical stage. Because of the mass, size and lack of video cameras I have never seen a rotary table centering instrument built using an alignment telescope. There may be such a configuration but I am unaware of it.)
This situation presents us with our first alignment task, the vertical slide needs adjustments in angle and position so the viewing instrument it carries moves parallel the axis of the rotary table as it is raised and lowered. To do this alignment optically we need an optical element sitting on the rotary table that has two separated but accessible centers of curvature along its optical axis. The BK7 element 10 mm thick shown in Fig. 1 is an example of an element that will work to produce the two centers of curvature.
Obviously, there will be a reflection 100 mm from the concave side. Looking into the concave side, the 175 mm convex surface looks like its center of curvature is about 249 mm above the concave surface giving 2 spots 149 mm apart, good enough to discern the angle of the vertical slide to about 1 second of arc when we measure the centers of curvature lateral position to +/- 1 µm precision.
Fig. 1 An optical element with 2 easily accessible conjugates to use to find the axis of a rotary table
You do not need a lens design program to do the calculation for a single element. If the left hand surface is R1 then its apparent center of curvature looking into R2 is R1optical = -R2(R1-t)/((R1-t)(n-1)-nR2) = 248.52 mm where n for BK7 at 640 nm is about 1.517. For more complex systems, a 1st design spreadsheet is very useful, and we will talk about some tricks to use with 1st order design in a later Chapter.
The two centers of curvature are easily viewed with either an autostigmatic microscope, or an autocollimator with an auxiliary objective of a suitable working distance. The element in Fig. 1 that you want to center is sitting in a seat that has adjustments for centering the seat relative to the table axis. With R1 sitting on the seat, the center of curvature of R1 will lie on the axis of the seat because a circle of any radius less than the radius of the sphere will lie on the surface of the sphere with the normals to the center of the circle passing thorough the center of curvature of the sphere as I have tried to indicate in Fig. 2. Thus, tapping the edge of the element in Fig. 1, or red circle in Fig. 2, will tilt the lens about the center of curvature of R1 and the center will never move off the axis of the seat. The idea in Fig. 2 is obvious but just because it is obvious is often overlooked. The concept is very important to centering.
Fig. 2 Center of curvature of sphere on seat lies on the axis of the seat
Of course, tilting the lens element in Fig. 1 by tapping will move the center of curvature of R2. To center R2 you must decenter the seat. The procedure to center the lens and seat is the same basic procedure as aligning an alignment telescope to an axis as discussed at the end of Chapter 3. In this case, the axis of the rotary table is our reference axis. We look at the apparent center of curvature of R1 since it is farthest from the point of rotation of the lens in the seat and tilt the lens by tapping on its edge until the reflected image is stationary as the table rotates. The image does not have to be centered on the crosshair in the viewing instrument, it just must be stationary as the table rotates to assure the image is on the axis of the table.
Then you move the viewing instrument on the vertical slide to the center of curvature of R2 and decenter the seat until the reflected image is stationary. Again, the image need not be centered, just stationary. This adjustment will invariably throw the center of curvature of R1 off the table axis. These steps are repeated iteratively by tilting the lens when focused at the center of R1 and decentering the seat when focused at the center of R2 to bring the two centers of curvature onto the table axis. The centering is finished when each of the two centers of curvature remain stationary as the table rotates.
Each adjustment will bring the centers closer until the centering meets some specification for the tilt and decenter of the lens where the element in Fig. 1 is an analogue of a lens. A skilled operator centering much the same lens element will quickly learn that you generally want to undershoot, or depanding on the lens shape, overshoot, the adjustment to bring the lens to complete centration more quickly. You must also keep track of the azimuth of the rotary table so you don’t undo the previous adjustment by tapping on the wrong side of the lens. Centration to bring the optical axis of the lens coincident with the axis of the rotary table is tedious but is as precise as the bearing of the rotary table and the patience of the operator. This is why this method of centering using a rotary table has been used for over a century. In a subsequent Chapter we will talk about an easier way to do centering without the need of a rotary table.
A decided short cut in the method is to know the seat is well centered to the rotary table axis in the first place as is obvious from Fig. 2. Then the only adjustment to the lens is tilting by tapping on the edge while looking at the optical center of curvature of R1. Effectively, having the seat centered orthogonalizes the centering process in that the decentering is completed and only tilting is necessary to finish the job. This idea is the reverse of the bond ring method mentioned in Chapter 6. There the lens was cemented in the bond ring free of tilt so only decenter was left to correct.
It is still wise to check that the center of R2 is still centered before assuming the job is done until you have enough experience to be sure there is no need to check. The least burr or contamination on the seat will ruin the assumption that the seat is acting as if it were centered.
To recap the points of centering an optical axis of a lens element to a rotary table axis we need:
A fixed focus optical sensor capable of projecting a crosshair or point source of light and detecting the return image of the projected source
The sensor must be mounted on a vertical stage and centered above the axis of a rotary table
The two centers of curvature of the lens element must be accessible to the sensor by moving the sensor along the vertical stage and/or by changing the objective lens of the sensor
There must be means of adjusting the lens in both decenter and tilt
When the reflected images of both centers of curvature are stationary (within some acceptable tolerance) in the sensor as the table is rotated, the optical axis of the element is coaxial with the axis of the rotary table
If the lens in incapable of being both tilted and decentered, or only one center of curvature is accessible, it is impossible to fully center the lens unless the seat accepting a spherical surface of the lens is well centered to the axis of the rotary table and the center of curvature that is opposite the one sitting on the seat
Before finishing this Chapter I should say that we have been discussing centering a lens optically on a rotary table. Mechanical centering is possible and is often used particularly for larger diameter lenses. Reviewing this method will reinforce what has been said about optical alignment.
To center a lens on a rotary table the seat is first centered on the table using mechanical indicators to assure the seat runs true to the table as in Fig. 3.
Fig. 3 Mechanical center of a lens on a rotary table
On the left of Fig. 3 mechanical indicators are used to assure the seat is running concentric and perpendicular to the axis of the rotary table. When the lens is first set on the seat it is clearly tilted and decentered, so the indicator shows a tilt or wedge in the lens as the table rotates. However, the center of curvature of the surface on the seat lies on the axis of the rotary table. As the lens is centered it rotates about the center of curvature until the indicator on the edge of the lens shows no height variation as the table and lens rotate. Then the center of curvature of the upper lens surface also lies on the axis of the table. Note the tip of the indicator remains the same height from the seat as the table rotates.
This also illustrates how bell cup centering machines work where lenses are made, see Fig. 4. The surfaces of the lens are squeezed between well centered cups to force the axis of the lens onto the axis of the cups. Obviously, this technique works best for steep bi-convex lenses and least well for meniscus lenses. The good news is that centering is most critical for lenses with considerable power and less so for weak lenses. The same holds true for centering in general but these are cases where the centering between pairs of weak lenses is critical because the wavefronts between the lenses have substantial spherical aberration.
Fig. 4 A screen shot from OptiPro’s website of bell centering cups seen squeezing a lens in the right hand photo
(I am not trying to play favorites here. Many optical machine manufacturers make similar machines. This company just had a picture that perfectly illustrates the point I was making.)
In this Chapter we will discuss the centering of a single center of curvature of a lens in a cell sitting on a rotary table that creates a reference axis. This discussion describes the traditional method of centering a lens in a cell. While this does not sound like an ambitious goal, the ideas presented here set the context for all the alignment topics to follow. We will get into much more complicated situations later, but it makes sense to walk before we run so that we understand the principals involved.
A rotary table was used for centering long before anyone heard of optics. Rotary tables, lathes or bearings are used to make objects that have no variation in the normal distance from the axis of the table as the table is rotated. If there is a method of measuring the height variation, we say the object is round and centered when no variation in height is observed normal to the axis of rotation by a fixed indicator or measuring device.
Notice there are two assumptions, one implicit; the object must be round, no azimuthal height variation and implicitly, the bearing true. The proper combination of a non-round object and poor bearing behavior can make it look like the object is centered. We will assume we have a perfect bearing so that any motion observed that is synchronous with the table rotation indicates decentration. This also implies that our centering will never be better than the bearing precision in tilt and decenter.
Using a rotary table introduces a practical constraint. In most cases the axis of the table is vertical so that gravity works for us. There are many examples where this is not the case, but for centering during the assembly of optics it is almost universally the case. The optical instrument, or sensor, viewing the optics being centered is mounted on a vertical slide centered on the axis of the rotary table. A scale to measure the height of the sensor that has a resolution consistent with the precision of the spacing of the lenses in the cell completes the hardware as shown schematically in Fig. 1
Fig. 1 Schematic diagram of a rotary table lens centering apparatus
The viewing, or sensing, instrument projects a point source of light, or an illuminated reticle, toward the lens being centered. When the projected source is at the center of curvature the light is normally incident on the upper concave surface and reflects back to the focal plane of the sensor and into an eyepiece or onto a monitor screen for viewing. As the rotary table revolves the reflected image will precess synchronously with the table unless the center of curvature is precisely aligned to the rotational axis of the table. The reflected image may not lie on the axis of the sensing instrument, but this only means the sensor in not well aligned to the axis of the table. If the image is stationary, it is on the axis of the rotary table. Fig. 2 helps with this explanation.
Fig. 2 Four possible alignment situations as the rotary table rotates the lens in Fig. 1
(The purple cross is the origin of coordinates within the sensor field of view, the black outline)
In Fig. 2a the center of curvature of the lens is not on the axis of rotation of the table nor is the sensor centered with respect to the table. In Fig. 2b the center of curvature is not centered on the table axis but the table axis is aligned with the coordinate origin of the sensor. In Fig. 2c the lens center of curvature is centered on the table axis because there is no motion as the table rotates but the axis of the table is not centered with the sensor origin, while in Fig. 2d the lens is centered and the table axis is centered with the sensor. In either case 2c or 2d the lens is centered with respect to the rotary table axis and that is all that is necessary for this one conjugate to be perfectly centered. That the reflected image does not lie on the center of the sensor has no effect on the centration of this surface of the lens.
When the reflected image precesses with the table, then the center of curvature does not lie on the axis of the rotary table as in Fig. 3a. There are 2 ways to move the center of curvature onto the table axis. Referring to Fig. 3b, we can decenter the cell and lens pair until the image is still. Alternatively, we can rotate the lens about the center of curvature of the surface sitting on the seat as in Fig 3c and decenter the cell to keep the lens from interfering with the cell. Either method, or a combination of the two will center the image, but this does not mean the optical axis of the lens (magenta dotted line) is concentric with the axis of the table, only that the center of curvature of the upper surface is coincident with the axis of the table at a single point. Only when the cell is centered, and the lens rotated about the center of curvature of the surface against the seat are the 2 axes coincident. There must be sufficient clearance between cell and lens to make up for centration errors during edging the lens or the lens hits the cell, 2d.
Fig. 3 With center of curvature not on the table axis (a), and on the axis (b, c and d)
Since we are only looking at one reflection for the moment, is there a way of completely centering the lens in Fig. 1 in the sense that it is free of tilt and decenter relative to the rotary table axis? Fig. 2d suggests that if we break the alignment into two parts, the answer is yes. First, we get the seat centered to the table axis using either mechanical or optical means. The centered seat and cell are locked to the table and the lens inserted. Since the seat is centered on the axis of the table, the surface of the lens sitting on the seat must also be centered.
To finish centering the lens we must tap the edge of the lens to tilt it in the seat until the reflection from the center of curvature of the upper concave surface remains stationary as the table rotates. The axis of the lower surface remains on the axis due to the mechanical interface between the lens surface and the seat while the upper surface is free of tilt via the optical reflection.
As a bit of a sidebar, this technique that separates the operations of removing decenter separate from tilt orthogonalizes the centering. A derivative of this method is called mounting lenses in poker chips or bond rings. These mounts are accurately parallel with seats parallel to the outer surfaces. Jump ahead to Fig. 4a to see what a bond ring with a convex lens might look like. Notice that the convex surface center of curvature will always lie on the axis of the bond ring. Tilting about the center of curvature of that surface will bring the center of curvature on the axis of the bond ring if there is clearance.
Assuming the surface of the rotary table is precisely normal to the axis of rotation and the seat of the bond ring centered, the lens can then be inserted and made tilt free as above by using a reflection from the center of curvature of one or the other surfaces. The optical axis of the cemented bond ring/lens pair is then precisely normal to the parallel faces of the bond ring. If all the lenses in an assembly are prepared in the same way, then the whole assembly is aligned by simply removing decenter as each addition element is added to the assembly. This method is often used to assembly the lens elements in large lithographic systems, and a close analog is used to assemble microscope objectives.
Whenever precise centration is needed in a lens system the bond ring approach should be considered. It gives the necessary adjustment needed to correct for both tilt and decenter while not having the procedure of adjusting for one upset the other. The adjustments are separate and orthogonal. The method does involve extra metal parts, but these additions are generally more than offset by ease and precision of assembly. Also, a design may only have one critical interface as far as alignment goes. Use the bond ring idea only where it is necessary,
To finish up this discussion, look again at Fig. 3. When adding the second lens to the assembly we must assume the seat is not completely concentric to the rotary axis. This means that when we attempt to center the lens, we only have the option to tilt the lens to center it. There is nothing we can do about the decenter. This leaves the question of what is the best tilt for optimum performance of the lens system. Currently in our discussion there is nothing to do but tilt the lens until the center of curvature is stationary. In a later chapter we will show there may be a better option for best centering.
There are analogous situations where the only option is to decenter the lens as in Fig. 4b when the interface is a flat on the lens, the usual situation when the curve is concave. Here it is impossible to tilt the lens if the flat is not perpendicular to the optical axis of the lens. The only way to correct for centration is to decenter the lens. In Fig. 4b this will largely correct for the wedge error in edging but not completely since the axes of the cell and lens are not parallel. These two situations point out the connections between the tolerancing of the cell and the lens. If the mating surface to a convex spherical surface is a seat as in Fig. 4a then the seat concentricity limits the centering of the lens. If the lens sits on a flat seat as in Fig. 4b, then the lens must be precisely edged to assure the flat is perpendicular to the optical axis of the lens. The cell seat centration is not important, but it must be perpendicular to the cell axis.
Fig. 4a Lens with a convex surface sitting on a bond ring seat with the concave surface perfectly centered by tilting the lens in the seat, and Fig. 4b where the lens sits on a flat seat and decenter is the only possible centration correction which will not completely work because the axes are not parallel
With these thoughts about tolerances in mind let me recommend the recent book by Herman, Aikens and Youngworth, “Modern Optics Drawings: The ISO 10110 Companion” published by SPIE. Because the optical drawings on centering in Chapter 8 all have to do with making and inspecting lenses as mechanical parts, virtually all the discussion is about mechanical methods of inspection and tolerancing.
This is just the area where many optical designers are unfamiliar with the techniques used to inspect optical elements. The book has numerous examples of how the tolerances on the drawing relate to the mechanical centration properties of the lenses and the mechanical methods of measuring them.
Hopefully I have given you an introduction to optical methods of verifying these measurements by making slight changes to the concepts in Figs 3 and 4. This introduction is hardly complete and in later Chapters we could revisit how to optically verify that lens elements meet the tolerances on drawings. Again, I highly recommend the book on the ISO drawing standard. It is used internationally and most optical drawings these days follow the ISO standard.
