Hopkins number helps avoid costly surprises

An important step in the design of an optical system is to consider how difficult it will be to manufacture. An optical design can look great on paper, but a tolerance analysis may reveal that it will be very costly to build. Sensitivities are often overlooked until a problem is found after the lens system is assembled and the performance does not meet expectations. A simple metric for estimating the difficulty of putting together a lens is very useful. One such metric was discussed at a recent ASE Optics lunch meeting by one of my colleagues, Peter Emmel. Peter described what he always thought of as the “Hopkins number” from his days of working at Tropel. He referred to it as such because it originated from Robert E. Hopkins, who was a co-founder of Tropel and is considered by many to be the “father of optical engineering.”

The Hopkins number is equal to the number of resolvable spots in the image and can be used to gauge the difficulty associated with making a lens. A “resolvable spot” is loosely defined in this case and depends upon what is limiting the system performance. For example, it may be the Airy disk for a diffraction-limited system, the geometric spot for a system with aberrations, or the pixel size for a sensor-limited system. The value can be computed from the 2D image area or, as Hopkins preferred, a 1D slice of the image area based upon its largest dimension. For example, with a rectangular image it would be number of resolvable spots along the diagonal.

At Tropel, Hopkins found that this number usually correlated with difficulties encountered in producing lenses across the full range of lens types and applications. Systems that had worse than anticipated performance almost always turned out to have a large Hopkins number. According to Emmel, one benchmark value was 6,600. “That was the 1D number for a particular first-generation microlithography lens that Tropel designed and built for Bell Labs in the 1970s. Tropel succeeded where two other firms had tried and failed to build lenses for this then-new application.” People learned to be wary of systems that had such a large Hopkins number because difficulties would be virtually guaranteed. For comparison, a Zeiss Distagon camera lens (T* 3.5/60 CFi for example) which has a 1D Hopkins number on the order of 3,900 would be considered “moderately difficult”. A point-and-shoot digital camera might have a 1D Hopkins number of roughly 2,500 and would be considered “easy”.

In Emmel’s opinion, the most important use of this metric is in the planning and cost estimating stages of a new prototype project. It provides a first indication of the likely sensitivity of a system to manufacturing tolerances. “Best of all,” says Emmel, “it can usually be calculated directly from the specifications, before a design is even started, making it a measure of the degree of difficulty inherent in the application.” Recognizing that a system may be particularly sensitive at the outset is much better than being surprised by poor performance after hardware has been built.

Technically speaking, the Hopkins number is related to the space-bandwidth product (SBP). For an optical system the SBP is equal to the image size divided by the size of the Airy disk and is therefore directly related to the diffraction-limited resolution. According to Goodman, “The space-bandwidth product of a function is a measure of its complexity. The ability of an optical system to accurately handle inputs and outputs having large space-bandwidth products is a measure of performance, and is directly related to the quality of the system.” [Goodman, “Introduction to Fourier Optics,” 3rd ed., Section 2.4.2, (2005)] For a diffraction-limited system where the Airy disk size is driving the performance, the Hopkins number is equal to the space-bandwidth product. For more on the SBP, see for example:

• Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, "Optical systems with improved resolving power," Prog. Opt., 40, 271-341 (1999).
• W. Lohmann, R. G. Dorsch, D. Mendlo vic, Z. Zalevsky, and C. Ferreira, "Space-bandwidth product of optical signals and systems," J. Opt. Soc. Am. A 13, 470-473 (1996).

The Hopkins number is a very handy metric for quickly estimating the degree of difficulty of an optical system. A high value means a challenging project, but steps can be taken in lens design to reduce tolerance sensitivities and in mount design to mitigate their impact.

Tolerance analyses and innovative mount designs are an integral part of the engineering work done at ASE Optics to ensure manufacturability. For “Hopkins number appropriate” optical system design, give us a call!