Author: csadmin

ABSTRACT

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

1. INTRODUCTION

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

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

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

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

2. DEFINITION OF CENTERING

2.1 Optical axis of a lens element

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

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

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

2.2 An example lens system

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

3. LOCATING THE CENTERS OF CURVATURE OF THE LENS SURFACES

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

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

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

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

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

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

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

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

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

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

4. POINT SOURCE MICROSCOPE (PSM)

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

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

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

5. VIEWING BOTH CENTERS OF CURVATURE SIMULTANEOUSLY

5.1 Scheme for simultaneous viewing

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

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

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

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

5.2 Modification for viewing with two autostigmatic microscopes

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

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

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

6. SENSITIVITY OF CENTERING

6.1 Theoretical considerations

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

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

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

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

6.1 Practical considerations

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

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

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

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

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

7. CONCLUSIONS

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

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

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

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

REFERENCES

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

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

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

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

ABSTRACT

The concept of centering a precision, symmetric lens system using a high-quality rotary table and an auto-focusing test instrument are well known. Less well known are methods of finding convenient, or easily accessible, lens conjugates on which to focus while performing the centering operation. We introduce methods of finding suitable conjugates and centering configurations that lend themselves to practical centering solutions.

1. INTRODUCTION

The act of assembling optical systems is often overlooked until the time comes to actually put the hardware together. By then thinking about the problem is too late because the customer is screaming for the system and there is no time or money to do the job correctly. The results are obvious; those responsible for the actual putting glass in cells are working overtime under great pressure and often the system produces less than optimum performance. This paper hopes to address some of the issues of assembly that can be looked at ahead of time so that some of the data needed by the assembly personnel are available when they are needed.

Another aspect of the same scenario is that Optical Perspectives makes an autostigmatic microscope, the Point Source Microscope (PSM)1 , that is useful during assembly for alignment and centering of the optical elements. As a result numerous questions are raised about how the PSM can be used during assembly, and even more comments, about why it appears to them that the PSM will be of little use. These comments generally come from the optical design people who worry about the things they see when designing lenses but have little applicability during assembly.

One reason is happens is that the design people look at perfectly aligned systems on their computer screens and have to purposefully misalign the elements to see how this affects performance. Further, if something gets so misaligned the answers do not make sense, all they have to do is hit the undo button and start over with a smaller perturbation. Another aspect of this is the way many design programs tolerance systems; they do not perturb element by element but surface by surface, something that cannot be done in practice because one surface is physically tied to a second and that has practical consequences as to what can be done to fix certain problems.

The situation is just the opposite for the assembly personnel; the elements come to them about as misaligned as they can be. One of the first things they have to do is make sure they are putting the lens in with the correct surface first. Sometimes they are given elements that are too large to fit the mounts or do not meet drawing specifications but they have no way of knowing because there was no incoming inspection. Their job is to get the assembly together quickly with no finger prints or other obvious defects and have it meet system level specifications. We want to consider some of the aspects of thinking about assembly ahead of time that will make life easier for the assembly personnel and hopefully lead to a better product.

2. ALIGNMENT AND CENTERING ARE FIRST ORDER PROPERTIES

The first point to make clear is that alignment and centering are first order operations; aberrations do not make any difference. If some surface is creating an image that is too aberrated to center on or to find a best focus axial position, stop the surface down enough to get a good image. Any good optic will have a smooth, continuous surface so light reflecting from a small region of the surface is representative of the whole surface from an alignment viewpoint.

A good illustration of this is finding the vertex radius and off axis distance of an off-axis parabola. If the mirror is at all fast the image viewed at the center of curvature it will have enough coma that the two astigmatic foci are difficult to isolate definitively. By stopping down the surface the coma is reduced and the two line foci can be measured with sufficient precision that the tangential and sagittal radii can be determined. Once these are known they can be plugged into formulas for the other parameters of the segment².

The second point is that, in terms of tilts and decenters, lenses can be thought of as wedges or prisms from a first order view point. As in Fig. 1, if a lens is decentered, the ray between object and image (an axial ray) strikes a small wedge at an angle related to the decenter and radii of the surfaces. For tilts, it is as if the ray strikes a plane parallel plate and is displaced but not changed in angle.

Assume we have a lens with radii R1 and R2, center thickness t and index n. Then the displacement of a ray along the optical axis will be d = t*α*(n-1)/n where α is the angle of incidence, that is, the tilt of the lens in this case. If a lens is decentered, the slope of the surface where a ray along the optical axis hits it will be y/R so the total wedge due to both surfaces will be γ = y*(1/R1 – 1/R2) so the deviation of the ray exiting will be β = γ*(n-1). Obviously these are small angle approximations but if the angles were large the lens wouldn’t work anyway. Also note that tilted lenses displace the ray and decentered lenses tilt, or change the angle, of the ray.

3. SIMULATION OF ALIGNMENT AND CENTERING IN A DESIGN PROGRAM

We can simulate the tilt and decenter in a lens design program using “coordinate breaks” as they are called in Zemax, or the equivalent, in other codes. As opposed to tolerancing where every surface and spacing is moved independently, once a real lens is made the front surface is physically tied to the rear surface and the lens code should reflect this fact. The code should be set up as in Table 1.

The coordinate breaks (CB) outside of the lens, lines 1 and 5, affect the whole lens and can be used to decenter or tilt it as a whole. The CB on line 1 introduces the decenter of tilt while that on line 5 removes its effect. Notice that by a judicious choice of tilt and decenter, the first surface will remain normal to a centered ray while the ray will encounter a tilted second surface. It is easier to use just the decenter without tilt and specifically introduce wedge using CB’s 1 and 3. Then it is easy to know exactly how much wedge is introduced. Furthermore, this is the way centering is specified in the ISO 10110 – Part 6 optical drawing standard3 , the wedge in the second surface relative to the first. Similarly, one could use CB’s at 3 and 5 to keep an on-axis ray normal to the first surface.

Notice that for thicknesses, the distance between the CB and surface is zero on either side of surface 1. Then the ray travels t to surface 2 where the final CB is immediately after the surface. It is important to get the spacings correct.

4. OPTICAL AXIS DEFINITION

My definition of the optical axis of a lens is one for which there are no exceptions, and that fits operationally what we are trying to accomplish, centering and alignment. The optical axis of a lens is the line joining the centers of curvature of the two surfaces. It is hard to argue with this definition and we will be doing our alignment using centers of curvature. A first apparent exception is a plano-convex or concave lens but here the axis is the line that passes through the center of curvature of the curved side and is normal to the plano side. Also, it is easy to establish a normal to the plano side with a collimated beam of light.

It is important to remember that an axis is defined by 4 degrees of freedom, either two points each defined by two degrees of freedom, or one point and one angle, each defined by two degrees of freedom. This means that when locating a lens, all 4 degrees of freedom must be constrained. Of course there is an additional degree of freedom that must be specified, the axial distance from some other datum such as a lens surface.

As a corollary to the above, note that a single spherical surface has no axis. It is defined by a point and a radius. A surface may appear to have an axis defined by the edge but this is mechanical and only defines whether a surface will be illuminated uniformly. The edge may be how the surface is mounted but does not define an optical axis.

5. CALCULATION OF CENTERS OF CURVATURE AND CONJUGATES

Now that we have settled on definitions we can get down to the heart of the matter, finding the lens conjugates. First assume we have a lens defined by R1 = 10, R2 = -6, t = 3 and n = 1.5. Assume, too, that we have a converging cone of light we are using to probe for the conjugates and that when the probe is focused on the vertex of R1 our height scale is set to 0 (see Fig. 2, top). The probe can be almost any device that has a real focus and it is helpful to have a reasonably long working distance as this gives more flexibility in terms of the ease of measuring surfaces. An autostigmatic microscope such as the Point Source Microscope is one example, but an interferometer and transmission sphere, or an alignment telescope (for centering only), could be used just as well.

While we are doing first order, or paraxial, optics, we also want to have axial sensitivity to best focus on conjugates. This means that it is good to have a fairly fast beam of light coming from the probe. A 0.1 NA cone or somewhat faster is about right to give good sensitivity to focus. Fig. 2 shows the 5 configurations described below.

5.1 The radius of curvature of the front surface

Assuming the probe lens used to produce a point source of light has sufficient working distance, the probe is moved from being focused on the vertex of the front surface toward the center of curvature of R1 . When the focus reaches the center of curvature a spot will appear and then shrink to a minimum diameter at best focus on the center of curvature. The distance the probe moved from vertex to center of curvature is the radius of the front surface. This radius is known without any further calculation and is generally the farthest the probe has to move in the positive direction.

When focused on the vertex a Cat’s eye reflection is gotten and the return spot of light will not moved laterally as the surface, or microscope, is moved laterally. The spot may vignette and loose brightness but does not move. This is different from being focused at the center of curvature. Here, a lateral movement of the surface or microscope will cause the spot to move return laterally.

5.2 Optical center thickness

If the microscope is move toward the lens from being focused on the vertex the first return spot is usually the Cat’s eye reflection off the rear vertex and the distance moved gives the optical, t0, as opposed to the physical center thickness, t. It depends on R1, n and t. We find from paraxial optics that

As seen in the third picture in Fig. 2 the fairly steep ray going into the lens is decreased in slope so it reaches the rear vertex before the projection of the incoming ray reaches the opposite vertex. Again, this will be the usual case but can change markedly depending on the radii and glass type. Watch signs carefully while doing the calculations.

5.3 Rear surface center of curvature

For a positive lens the rear surface center of curvature will generally lie outside the lens, or on the negative side of zero the way we have defined things. Here the radius of the first surface, index and thickness as well as the rear surface radius will all affect the apparent rear surface center of curvature. As shown in Fig. 1 the rays entering the first surface are bent toward the axis so the center of curvature is closer to the first surface than the physical center of curvature.

In this case we have

where R2 is negative.

5.4 Back focal length

Calculating the back focal length is particularly useful for the case when the first surface radius is too long for the working distance of the probe being used. In this case a plane mirror is placed behind the lens being centered or investigated and a focus will be found outside the first surface vertex for a positive lens as in the lower panel of Fig. 2. If the other conjugates can be measured easily, the bfl measurement is a good cross check because as one moves along the axis, the conjugate foci are not labeled as to where they are coming from and it is easy to make a mistake. Having a way of cross checking is very useful.

For the bfl we have

You can do the first order ray trace or check reference² for the derivation.

6. FINDING CONJUGATES USING A LENS DESIGN PROGRAM

Another way of doing what has been described above is to use a lens design code. Fig. 3 shows the five configurations above in the Multi-configuration editor in Zemax. The IGNR operand tells the program to ignore that surface for a particular configuration. The code will optimize to find the correct spacing of surface 2 to find the conjugates we are looking for on line 3 of the Multi-Configuration Editor. Remember that a finite aperture is needed to get enough rays through to optimize but a large aperture will give spacings that correct aberrations, not paraxial spacings. One way to check if the aperture is small enough is the compare the optimization value of the bfl with the General data in the prescription. The general data are paraxial calculations and will directly calculate the bfl from a paraxial ray trace.

7. DETECTING CENTERS OF CURVATURE AND CONJUGATES

Assuming now that the lens is sitting on a seat ready for centration on a rotary bearing, conjugates that are vertices will produce Cat’s eye reflections and will not move even if the lens is decentered. The Cat’s eye reflections are useful for setting crosshairs in x and y as to where a center of curvature return should appear on the video screen because the Cat’s eye falls on the optical axis of the test device or probe.

A reflection from a center of curvature will move whenever the lens is rotated on a rotary table unless the lens is perfectly centered. The center of the rotation of the spot depends on whether the microscope objective is lying on the mechanical axis of the rotary table but has nothing to do with how well centered the lens is. It is convenient to have the return from a center of curvature rotate about the center of the screen but it is not necessary. If the return spot moves the lens is not centered, or you have a rotary table with too much wobble to be useful.

Since we have said the optical axis is the line joining the centers of curvature it makes sense to center using these conjugates. Sometimes one or the other is inconvenient to reach and another such as the bfl is a better choice for one of the conjugates.

To give a general example we use the example lens from above and show how by adjusting the decenter of the lower seat and the tilt of the whole lens, the lens can be centered. As shown in Fig. 4, the rear surface sits on the seat and in general is both tilted and decentered to start with. With the probe focused on the center of curvature of the rear surface, the seat is centered until the focused spot does not move. Then the probe is moved toward the lens down to the front surface center of curvature and the lens slid in the seat until the focus spot does not move. Notice the lens is tilting around the center of curvature of the rear surface and this is why the seat must be centered first and then the lens is tilted. If it is done the other way around the process of centering diverges.

The centering should be complete in these two steps but it is best to go back and double check the upper or rear surface center of curvature to make sure nothing moved while centering the second surface. Also, if there is substantial tilt of the lens to begin with the center of curvature of the rear surface will not be physically where it appears to be optically because of refraction at the first surface. This may also call for iteration to make sure the lens is completely centered. Once both centers of curvature are stationary, or centered within the tolerance, the lens may be cemented.

7. CONCLUSION

We have pointed out that alignment and centering operations deal with the first order properties of lenses and mirrors. If aberrations interfere with obtaining good conjugate data the surfaces may be stopped down to obtain good images for alignment. Once a definition for the optical axis is given it is easy to find the conjugates for a lens, and then use these and a precision rotary table to center lenses in a cell. Following a step by step procedure it is not difficult to get centers of curvature to lie on an axis to a few micrometers TIR.

REFERENCES

[1] Optical Perspectives Group, LLC

[2] Smith, W., [Modern Optical Engineering, 3rd ed.], McGraw –Hill, New York, p. 485 and 40 (2000).

[3] Available from https://webstore.ansi.org/