OPTICAL DESIGN: Understanding and predicting self-weight distortion of lens elements
FRANK DeWITT IV and GEORGE NADORFF
The distortion of optical surfaces as a result of gravity loading is an issue that faces many lens designers, mechanical engineers, and metrologists. While this effect has often been associated with very large optics such as those found in ground-based telescopes, the constant push toward higher performance and greater accuracy requires designers to consider gravity in the tolerancing and design of smaller optical assemblies as well.
Lens designers and mechanical engineers often use "rules of thumb" to decide how thick a lens element or mirror needs to be to adequately resist the effects of gravity. Although these guidelines can be useful, it is important to understand their limitations and decide when more analysis is required. Finite element analysis (FEA) provides the engineer with a powerful tool for predicting the effects of gravity on an optical element. Care should be taken to ensure that the boundary conditions are chosen appropriately to accurately model the interaction between the optical surface and the mechanics that will hold it in place.
Self-weight distortion of an edge-supported cylindrical plate
For a circular plate under the effects of a uniform load, this load is the result of the weight of the part itself. In this orientation, an element will sag and its upper and lower surfaces will become distorted. The analogous real-world situation would be an optical window or a thin mirror sitting freely on a compliant ring near the outermost edge.1
The scale of the part, its thickness, and the specific stiffness of its material will all affect the extent to which the part will deflect. For the case of a cylindrical or circular plate, the amount of deflection can be predicted using the closed-form solutions found in Roarks Formulas for Stress and Strain.2 The central deflection (yc) of the part is given by:
Here, t = plate thickness, n = Poisson's ratio, E = Young's modulus, a = radius of the plate, r = material density, and g = acceleration due to gravity.
Using the equation given for yc, we can explore the effects of part thickness, part scale, and specific density on the self-weight deflection of a circular plate. For purposes of illustration, we consider a window made of Schott F5 optical glass. It has an aspect ratio (the ratio of the diameter of the lens to its center thickness) of 8:1, with a diameter of 4 in. and a thickness of 0.5 in. The parameter yc (converted to waves at 633 nm) can be plotted as a function of scale factor by independently varying the thickness, specific density, and part scale from 0.5x to 2x the original values.
As expected, the part thickness has a significant effect on its self-weight deflection. In addition, the scale of the part has a similar but opposite effect. Therefore, for the F5 disk (even as the aspect ratio of 8:1 is maintained), if the dimensions are uniformly scaled either up or down, the self-weight deflection is dramatically changed. While not as severe, specific density also affects the magnitude of self-weight deflection. Specific densities of common optical glasses can range from 10.81 x 106 (SF6) to 32.27 x 106 (BK7).
Depending on what level of sag deformation of an optical element is tolerable, the analysis suggests that "rules of thumb" using element aspect ratios alone are not sufficient to guarantee as-used performance. At a minimum, they are only relevant for optics of similar diameters and similar specific densities, and should be used with caution. Material type and diameter must also be considered.
The effect of imperfect parts
Our discussion to this point has centered on the self-weight deflection of an element that is supported uniformly and circumferentially around its edge. An analogous mounting method is to support the element with a compliant ring, or to mount the element with a uniform adhesive bond around the circumference of the optical element. However, in many cases, it is desirable to allow the element to rest on a hard seat to establish the axial position of the element.3 This hard seat will interact with the element very differently than the ideal seat that is assumed in the closed-form solutions. A thin lens of large diameter resting on a rigid seat may distort nonuniformly as gravity pulls the lens against the irregularities of that seat. The machined surface of the seat is often highly irregular relative to the near-perfect form of the optical surface that will be resting on it. These small irregularities can cause an element to distort to a higher degree than it would if placed on a compliant ring.
A common form of an irregular lens seat might be a seat that is simply out of round. Many materials will "spring" or change from circular to elliptical in shape during machining. While this effect is small—considering that the optical surface contacting the lens seat is almost perfectly spherical—even a seat that is out of round by only micrometers can be problematic. In this case, a spherical surface will contact primarily two points on the minor axis of the ellipse, which will act as a fulcrum along which the element will distort.
For a specific meniscus positive lens element resting on a perfectly circular seat, a FEA model can be created. The resulting peak-to-valley (P-V) distortion of the concave surface is 6.7 nm. In addition, the contours of departure from the pre-stressed condition can be overlaid on the lens. The lens element is constrained to rest on only two points oriented 180° apart and separated by a distance equal to the seat diameter. The surface shows a "saddle" shape with P-V distortion of 47.5 nm. Therefore, analysis of the distortion with two-point support is approximately seven times larger than the distortion predicted using FEA with a perfectly circular seat.
A compromise solution is to intentionally support the element on three equally spaced points around the circumference of the lens seat.
This interface, though not as ideal as a perfectly uniform seat, avoids over-constraint of the lens surface while being more predictable and less affected by manufacturing tolerances. In this case, the P-V distortion is 18.35 nm, or approximately only three times larger than the distortion predicted for the compromise solution with a perfectly circular seat. In general, an element that is supported on three points (equally spaced around the contact diameter) represents the best possible condition for successful mounting against a rigid seat.
Determining a starting point
We can combine the simplicity of the closed form solution and the FEA modeling of imperfect parts by reorganizing the equation for yc to solve for t and simultaneously including a factor "S" that will be determined empirically.
This equation allows us to find a minimum thickness that would result in a given distortion due to gravity loading. To apply this equation to a lens, we note that the induced irregularities on the lens element—when sitting on an elliptical seat—were roughly seven times the irregularity observed when the element was supported on a circular seat. Therefore, the value of S would equal 7, and we can then solve for a thickness value that will limit the self-weight deflection to a value, yc, that we chose. If the boundary conditions, material properties, or element shape changes, then the value of S will need to be redetermined by another FEA run.
Such an approach can be helpful during the optical-design phase of a project as a starting point for a lens thickness or material. The further a lens shape departs from a plane-parallel plate, the less accurate this prediction becomes. A more thorough FEA analysis will be required to verify the final lens form.
To test this concept, a study was done that applied the closed-form solution for yc to three basic lens shapes and compared the results with FEA predictions.4 The three lens shapes were plano-convex, plano-concave, and meniscus, and were designed to have radii of 2.2 times the diameter. This radius-of-curvature to edge-diameter ratio was chosen to represent a fairly standard lens shape without extreme curvatures. The parts were modeled to have equal cross-sectional average thicknesses. The closed-form solution was applied by assigning the average cross-sectional thickness for the value "t." The results were within 30% of what FEA predicted for the same element shapes. The deflection predicted by the closed-form solution of a plano-plano part, of the same diameter, material, and thickness, resulted in a 3% agreement with the FEA.
When the approximate nature of the technique is properly accounted for, this approach can be particularly useful in determining how much thicker or thinner a part must be in order to meet a self-weight deflection requirement. When S has been predetermined for a given case, the equation can be quickly used to determine the effect a given thickness change will have on the self-weight deflection of the part.
Given the increasing demands placed on allowable surface irregularity, attention must be paid to self-weight deflection of optics and the resulting optical surface distortions. In some cases, it may be appropriate to simulate and analyze the deflection of an optical element as resting on a perfect lens seat. In other cases, the boundary conditions must be adjusted to account for the likely contact points of the optical element on surrounding metals. In cases where the contact is not well distributed, the distortions are not only greater, but also of a more disruptive nature since they are generally no longer circularly symmetric. The closed-form approximation for plate deflection can be a useful tool for understanding the contribution of the various drivers. Material properties and part dimensions need to be considered. Finite element techniques, coupled with experimentation, are likely to be required when the boundary conditions or geometries of an optical element become more complex.
REFERENCES
1. D. Vukobratovich, Introduction to Opto-Mechanical Design, Course Notes (1993).
2. W. Young and R. Budynas, Roark's Formulas for Stress and Strain, McGraw-Hill (2002).
3. F. DeWitt IV and G. Nadorff, Proc. SPIE 5867, San Diego, CA (2005).
4. F. DeWitt IV, G. Nadorff, and M. Naradikian, Proc. SPIE 6288, San Diego, CA (2006).
Frank DeWitt IV is principal optomechanical engineer and George Nadorff is principal optical engineer in the Melles Griot Optics Group, 55 Science Parkway, Rochester, NY 14620; e-mail: [email protected]; www.mellesgriot.com.