#DevOptical Part 31: Cell Mounting of Lenses

Mounting lenses is a vast topic and entire books have been written on it. For newcomers in optics, the field of optomechanics is often ignored but it is an essential part that will make or break your designs.

Because I can’t cover the complete topic in a blog post, I will focus on important aspects that will put you up to speed in the domain. I will cover only the most important mounting methods for small lenses and skip those that are not in use or that would deliver sub-optimal performances. In this post, I will focus on cell mounting of lenses. In a second post, I will discuss barrel mounting which is a method that can be seen as a generalization of the cell mounting strategy.

But before we delve into the intricate details of cell mounting, it might be important to understand why “standard” mounting is not desirable – or, more specifically, when it is not desirable.

When assembling breadboards using off-the-shelf components, you will encounter two main mounting strategies: diameter fitting, and tube mounting. In the former solution, the lens is inserted into a mount and secure by its outer diameter using a set screw. In the latter, the lens is inserted in a threaded tube and secured using retainer ring. Example of diameter fitting at Thorlabs are the CP35-type mounts and example of the tube mounting are the CP33-type mounts which are both represented in Figure 1.

If you have read my former post on the [»] effect of tilts and decentering in lens systems, you already know that holding a lens by its diameter is a recipe for catastrophic optical performances. Due to the manufacturing process of lenses, there is no guarantee that the axis defined by the outer diameter of the lens perfectly matches the natural optical axis of the lens formed by the line joining its two centers of curvature. When you slant or decenter a lens optical axis in regards to the light rays, you generate asymmetrical aberrations such as coma and astigmatism which degrades image quality.

But holding the lens by its outer diameter also has consequences in the thermal stability of your system. When increasing the temperature, either during operations or during storage, the lens mount will typically grow faster than the lens because the thermal coefficients of expansion of metals are usually larger than those of glasses (to be checked on a per-case basis!). At some point, there will not be enough tension to hold the lens in place and the lens will become free to move, leading to potential misalignment or even fracture with small shocks. Even worse, when the temperature decreases, the lens mount will shrink faster than the lens. At first, the tension of the set-screw will increase but as the temperature keep decreasing, the gap between the mount and the lens might disappear and the metal will then start pushing extremely hard on the glass, generating massive Hoog forces. This will result first in a decrease of optical quality due to birefringence, and will end with a catastrophic failure of the lens itself (i.e. the lens will break!).

Finally, by clamping the lens using a set-screw, you generate a relatively uniform vertical stress throughout the lens. By deforming the glass lattice, even minutely, the material becomes anisotropic leading to different refractive indices in the direction of the stress and perpendicular to the stress. This effect, known as birefringence, is generally considered as being problematic for polarization-sensitive setup only, but the truth is that it affects all optical systems by creating astigmatism (different foci along the stress direction and its perpendicular).

These are the main reasons why you should never hold a lens by its diameter outside the scope of a laboratory breadboard that will not encounter temperature variation during its storage or operations.

Tube mounting of lenses solves many of these issues: pressure is applied axially through the retainer rings, leading to less problematic birefringence, a sufficient gap between the lens outer diameter and the mount is usually present and, when being mounted, the lens will spontaneously align to its natural optical axis.

Figure 2 illustrates the process by which the lens aligns with its natural optical axis during mounting (I call here “natural optical axis” the axis formed by joining the two center of curvature of the lens, independently of the lens mechanical shape after manufacturing). As pressure is uniformly applied by the retainer ring, the lens will rotate in the mount until the center of curvature of both of its faces aligns with the axis of pressure generated by the mount and the retainer ring.

The tube mounting strategy looks more promising, but some caveats need to be identified. While it is true the lens will align with the pressure axis generated by the mount and the retainer ring, there is currently no control on where the axes are in regard to each other’s but also in regard to the mount itself. The cell-mounting of lenses method answers this question by introducing tolerances on the different elements.

Figure 3 shows a typical cell-mounted lens. While it is very similar to the tube-mounted version, we see that each surface is tightly controlled through a series of tolerances. Multiple cells can then be inserted into a toleranced barrel (typically G6 inner diameter) and have all their natural optical axis matches each other, as well as getting precise control on the inter-distances of lenses.

The being said, not all lenses will center using this process. According to Paul Yoder in “Fundamental of Optomechanics”, a lens will center only if the following condition is met

where the y_c and R are the contact heights and radii of curvatures of the lens, and µ_G is the glass-metal friction coefficient.

Still following Yoder, uncoated lenses have friction coefficients around µ_G≈0.15 but coated lens can have higher friction coefficients (MgF₂ coating on black anodized aluminum is reported to have µ_G≈0.38). Concave surfaces have negative radii of curvature in this formula, making meniscus lenses difficult to self-center. It is therefore important to always have a look at this formula before deciding to mount the lens using this technique.

I did not find the mathematical derivation for this formula but y/R being the angle of the lens at height of contact, I assume this has something to do with a condition where normal pressure on the lens transforms to torque through the friction coefficient. If you have more information on this, please share them on the [∞] community board.

Let’s now review what are the degrees of freedom of our lens-mount system such that we understand the importance of each tolerance.

To locate our lens in space, we need 5 degrees of freedom for its XYZ and θ_xθ_y (θ_z is irrelevant because the lens is of symmetry of revolution). Among these 5 degrees of freedom, lateral XY and θ_xθ_y can be split between magnitude and direction amounts. Since the direction will be random and can change at each mounting/unmounting of the lens system, we care only about their respective magnitudes – leaving 3 quantities to identify: amount of Z displacement, amount of XY displacement and amount of θ_xθ_y tilt. I will rename these axial displacement, lateral displacement and tilt.

The mount-lens system can be analyzed in terms of interfaces. We can list:

(1) Interface between the lens and the cell, which is a circle;

(2) Interface between the lens and the retainer ring, which is also a circle;

(3) Interface between the cell and its barrel via their respective diameters;

(4) Interface between the retainer ring and the mount via their respective diameters;

(5) Interface between adjacent cells via their back and front surfaces.

If we consider that the reference coordinates system is the axis defined by the outer diameter of the cell, we can state that:

(1) The tilt of the lens is driven by the perpendicularity of the lens-cell interface with that axis relative to the diameter of the circular interface;

(2) The lateral displacement of the lens is driven by the concentricity of the lens-cell interface with the reference axis;

(3) The axial displacement of the lens is driven by distance between the lens-cell interface and the front surface of the cell.

Furthermore, when mounting multiple cells in a barrel, we also need to account for the tilt of the cell itself and the overall thicknesses of each cell as they are stacked together.

The typical tolerances annotations required are summarized in Figure 4 for axial displacement, lateral displacement and tilt.

Identifying the tolerances to use for a given system is a back and forth process between optical design tolerance simulations and machinability of the cell. The value given in Figure 4 are a good start for common lenses and lenses systems but should be refined depending on the optical design sensitivity. As this goes beyond the scope of this post, I will leave it for later.

A few other tolerances still need to be discussed but their effect is less straightforward to analyze.

First, the lens-cell interface might not be a perfect circle due to the machining process. Irregularities of the interface circularity and diameter variations of the interface can produce both lateral and axial displacements. Lateral displacements will be driven by the circularity of the interface whereas axial displacements will be driven by diameter tolerances and the curvature of the lens (the flatter the lens, the less sensitive it becomes to diameter variations).

Second, it is important to carefully draft the tolerance for the retainer ring interface. If the retainer ring axis is not concentric with the lens-cell interface axis, the system becomes over-constrained, and it becomes difficult to know which from the cell or the retainer ring becomes the driving mechanism for the alignment of the lens natural optical axis. To avoid mismatches, the thread pitch diameter must be concentric with the cell main axis. We can also make the system more resilient by using a loose-fit between the retainer ring and the cell such that the retainer ring can adjust its lateral position to avoid over-constraining the system. Some textbooks will also recommend a specific angle for the threads, but it would require specific tools at the machining workshop and will either be ignored or reflected on the price of the parts and lead time.

Additionally, as for general prescription, it is recommended to keep the front contact point at roughly the same height as the back contact point to avoid bending moments on the lens. Lenses are very sensitive to shear stress so keep in mind to have the front aperture about the size of the inner diameter of the retainer ring.

Also, when mounting cells in barrel, it is recommended to avoid having the same metals in contact. I have had extremely bad experiences with aluminum-on-aluminum cell-barrel contacts, even with coating on them. Brass on aluminum is a very popular choice so I recommend using MS58 brass as default material for the cells and 6XXX or 7XXX series aluminum for the barrel.

Figure 5 illustrates different mounting scenarios that you will face during a typical design. I would recommend putting the most curved surface on the cell side although I do not have evidence to show that it gives higher precision. Using concave surfaces is possible, but you need to introduce a lip in either the mount or the retainer, as can be seen in Figure 5. When mounting a very thick lens, such that the cell becomes relatively long, it is customary to introduce bearing surfaces to facilitate cell insertion in the barrel and avoid over-constrainment (see Figure 5). Finally, when pushing against a flat surface, remember that the lens will align relative to the perpendicular of that face – so check with your lens manufacturer what tolerances they can offer on the flat surface in regards to the optical axis.

One last tip that I can give when designing cell mounts is on lens protrusion from the cell. If possible, you should avoid having convex surfaces leaving the cell because it is very tempting to put the cell flat on a table – in which case you are very likely to damage your lens as shown in Figure 6. A gap of at least 0.5 mm between the lens vertex and the cell front or back face is a good starting point for your design. It is however not always possible to prevent the lens from protruding, typically when you need to stack two cells, one with a convex surface and the other one with a concave surface. If you let the lens protrude from the cell, always lay the cell on the safe side. The danger arises when the lens protrudes by a very small amount only because it will not be apparent at first sight, and you will therefore pay less attention when handling the mounted lens. My recommendation here is either not to let the lens protrude, or to have it protrude by such amount that it can easily be spotted during handling.

To assist you in designing your own mounts, I created a [∞] web application that includes default tolerances.

Up to now we essentially focused on mounting single lenses (e.g. for a simple telescope), but we did not discuss how to deal with lenses assemblies. While it is perfectly valid to stack multiple cells on top of each other, you may have trouble with lenses that are very close to each other. An alternative is to divide the cell into multiple parts such as shown in Figure 7. Th stack is now composed of what I call L-junctions and T-junctions, from the shape or their respective revolution profiles. The stack does not hold tight by itself, so it requires being mounted in a barrel with a lip at the front and a threaded section at the end to mount a thick retainer ring.

L/T/L stacks are very common in optics but lack some advanced features that single cell offers. We saw that, because of the machining tolerances, the alignment between the lens natural optical axis and the cell outer mechanical axis will never be exactly the same. I also said that optical simulations were required to determine what tolerances were acceptable for a given optical design. It happens that, for some very sensitive systems, the tolerances are still too tight to be obtained through conventional machining and mounting procedures. To address this problem, it is possible to mount the cell with its lens on a high-precision lathe that can adjust its spindle axis to the natural optical axis of the lens and to rectify the cell such that the mechanical axis and thicknesses correspond to the required ones down to a fraction of arcmin. The process is expensive and needs a very specific lathe that can locate the two centers of curvatures of the mounted lens with high-precision. When such methods are used, it is common to bond the lens in the cell using special adhesive to avoid the lens from getting loose (or be unmounted) which would screw up the complete rectification job in an irreversible way. Bonding lenses goes beyond the scope of this post as it requires a precise gap between the lens and the cell to accommodate thermal changes.

Still talking about thermal effects, note that our designs include an air gap between the lens and the cell to accommodate for thermal effects as well. The evolution of clearance between the lens and the cell with a temperature change ∆T is given by

with α_M the thermal coefficient of expansion of the metal, α_G the thermal coefficient of expansion of the glass, and D_G the outer diameter of the lens.

For a 1” N-BK7 lens mounted in a brass cell, the amount of radial clearance changes by about 0.15 µm/°C. To survive a -62°C storage from a 20°C mounting temperature (typical military requirements), a clearance of minimum 12 µm between the lens and the cell is therefore required. Adding default tolerances of an ISO 2768 fH drawing, I would recommend to leave a gap of minimum 0.3 mm between the lens and the cell mount to be safe.

Thermal load also modifies the axial distances. As the system cools down, the cell shrinks faster than the lens – leading to additional force being applied on it. Inversely, as the system heats up, the cell grows faster than the lens, loosening the retainer ring pressure and eventually creating some axial gap. While this goes beyond my expertise, Yoder gives very straightforward formula for simple lenses in a cell mount like we are dealing with.

The load (in Newtons) associated with a thermal change ∆T is

with

where

and t_c the cell wall thickness at the rim, t_E the lens edge thickness at contact height y_c, E_M the young modulus of the metal, and E_G the young modulus of the glass.

As an order of magnitude, our typical 1” N-BK7 lens in a brass cell would have a temperature sensitivity of -50 N/°C. Surviving a +62°C storage environment from a +20°C ambient temperature would require a preload of at least 2kN (torque of 5 Nm on the retainer ring). At -72°C, the additional preload would be another 2.5 kN.

For doublet lenses, the formula needs to be adapted to

where the various subscripts stand for the first and second lens respectively.

For L/T/L systems, the formula is further adapted to

with ‘s’ the subscript for the T junction and

and

The formula expands similarly with more T junctions.

Similar to thermal loads, any shocks or vibration will generate a load that might overpass the force exercised by the retainer ring. Yoder gives

with W the mass of the optics and A the maximum acceleration factor expressed in g’s.

The total preload applied by the retainer ring should therefore be larger than P_shock+P_thermal but small enough such as not to rupture the glass. Still following Yoder, the stress induced by a given preload is

where

and

where P is the preload, D₁ is twice the radius of curvature of the lens surface in contact with the mount, D₂ is the radius of curvature of the edge that contact the lens surface, ν_G is the Poisson ratio of the glass and ν_M is the Poisson ratio of the metal.

For sharp corners, a value of D₂=0.10 mm is frequently used. Other geometries are possible, such as tangent surfaces, but are more difficult to machine reliably (a maximum 2° variation for the tangent angle is recommended by Yoder). When using tangential contact points, K₁ decays to 1/D₁.

The amount of torque required for the retainer ring to produce a given amount of preload is

with D_R the pitch diameter of the thread of the retainer ring, µ_M the metal-metal friction coefficient, and µ_G the metal-glass friction coefficient.

Experiments done by Yoder indicates that µ_M≈0.19 and µ_G≈0.15 yielding

For lenses with high preload, it is important to also check that the pitch of the thread is large enough to withstand the stress. Still following Yoder we have

with D_T the thread diameter.

For a 1”-40 retainer 6061 aluminum in a MS58 brass thread, the maximum preload is on the order of 25 Nm to avoid going beyond the tensile strength of aluminum. A safety factor of ~5 is typically recommended yielding a maximum torque of 5 Nm. Note that the maximum preload should be used for this computation, not only the preload at ambient temperature.

Our 1” N-BK7 lens example clearly illustrates the problem of thermal loads on optomechanical assembly. When the stress induced by the storage temperature goes beyond the limit of the lens, it becomes mandatory to select materials with coefficients of thermal expansion that better match those of the lens(es). Stainless steel and titanium are popular choices for such applications.

Evaluating the exact amount of a stress a lens can withstand before failure is difficult, according to [∞] Schott themselves. I have run across multiple strategies to evaluate a glass toughness to fracture, including Weibull statistics modeling, but none of the method seemed very straightforward and required many inputs. Also, as mentioned in Schott’s document, post-processing on the glass can alter its resistance to stress. The value of 345 MPa is sometimes found as a “safe” threshold, but it gives extremely low torque values that are not very practical on the field. The only thing that seems to be certain is that some glasses are more brittle than others, such as calcium fluoride. When facing high preloads due to environmental requirements, switching materials for the cell is probably the safest option of all.

This concludes our post on cell mounting of lenses! You should now be an expert in the domain :)

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