**Key specifications and their impact on performance**

## Introduction

Any spherical optical surface, even those perfectly designed and manufactured, will exhibit spherical aberration. Due to this inherent defect in spherical surfaces, incident light rays focus at different points when forming an image, creating blur. Aspherical lenses were designed to correct this defect. (Figure 1).

Aspherical lenses can offer an improved spot size, several orders of magnitude smaller than spherical lenses, virtually eliminating blur and significantly improving image quality. Aspherical lens elements also enable designers to create higher-efficiency systems while maintaining good image quality in multi-element assemblies. In optical systems, several spherical elements can be replaced by a single aspherical lens, reducing size and weight without compromising performance. In recent years, manufacturing methods have continued to improve, and aspherical lenses have become a staple of modern optical design.

Any lens whose surface is not spherical can be called aspherical; however, for manufacturing reasons, most aspherical lenses are rotationally symmetrical lenses whose radius of curvature varies from the center to the edge. This geometry poses unique challenges not found in traditional lens manufacture. A spherical lens is defined by a single radius of curvature and can be ground and polished by a tool larger than the component, working the entire surface at once. In contrast, the continuously variable radius of curvature of an aspherical lens requires under-aperture polishing by tools small enough to create different curvatures localized at different points on the surface.

Aspherical lenses are both fundamentally more complex to design and manufacture than spherical components, so it's important to know their individual specifications and what this means for performance.

### Specifications

An aspherical surface is usually described in terms of sag, which can be thought of as the deviation of a plane from its apex. The equation is given below:

** Z**(s) is the offset of the surface from the vertex at a radial distance of

**from the optical axis. The parameter**

*s***is the curvature (which corresponds to the inverse of the radius of curvature at the vertex) and**

*C***is defined as the conic constant. The terms A**

*k*_{4}, A

_{6}and A

_{8}are called aspherical coefficients of 4

^{ème}, 6

^{ème}and 8

^{ème}order. Figure 2 illustrates the comparison between an aspherical and a spherical surface.

The aspheric surface described by the equation above represents the ideal shape; the aim of manufacturing is to get as close to it as possible. Inevitably, there will be a deviation from the ideal surface profile: this is known as the aspheric figure error, or **surface irregularity**. This is calculated by subtracting the ideal surface from the manufactured surface using software, and analyzing the residual deviation. This specification is often quoted as a peak-to-valley (P-V) value, which represents the difference between the points of maximum and minimum deviation. However, this value can be misleading, as it does not specify the number of peaks and valleys present on the optical surface. A more reliable measure of surface irregularity is the Root Mean Square Deviation (RMSD), which examines the absolute difference from the ideal surface at several points and calculates an average value for the entire optic. This value can vary from a few microns for commercial-quality optics to a few tenths of a micron for high-precision optics.

Although surface irregularity gives a good indication of lens performance, a significant amount of information is still missing. If we consider the entire optical surface, the deviation from the ideal shape at any given point is not constant. In addition, the under-aperture grinding and polishing techniques used in aspheric lens manufacture can create repetitive patterns and structures. *in* the surface irregularity profile, known as mean spatial frequencies. Another key specification is the irregularity slope or **slope tolerance**. This value sets an upper limit to the rate of change of the aspherical figure error, describing how quickly the deviation from the ideal shape can change within a given window. Typical values range from 1 µm/mm for commercial quality to 0.15 µm/mm for high precision. The window size is an important part of the specification and should be chosen to be less than the wavelength of the targeted mean spatial frequency, but large enough to avoid counting higher frequency variations, such as surface roughness or instrument noise.

### Performance considerations

All optical systems have a theoretical performance limit, called the diffraction limit. The Strehl ratio is a specification for comparing the actual performance of an optical system with its diffraction-limited performance. For aspherical lenses and other focusing optics, the Strehl ratio is defined as the ratio between the maximum focal point irradiance of the manufactured optic and the maximum diffraction-limited irradiance.^{1}. The industry standard threshold above which a lens is classified as "diffraction limited" is a Strehl ratio of 0.8.

The Strehl ratio can also be related to the RMS transmitted wavefront error using the following approximation, in which the RMS wavefront error is represented in waves^{2}. This approximation is valid for transmitted wavefront error values < 0.1 waves.

The Strehl ratio of an optic is highly dependent on its surface accuracy, which can be quantified in terms of**surface irregularity** and **slope tolerance**both described in the previous section. First, consider the spatial frequency of the figure error. When surface irregularity is modeled as a rotationally symmetric cosine function, we can explore the resulting Strehl ratio as a function of RMS surface irregularity for a variety of cosine periods (*Figure 3* and *Figure 4*).

Here, the key factor is not the cosine period in mm, but the *number of periods on the lens aperture*. For a given under-aperture tool used in the manufacture of aspherical lenses, small-diameter aspherical lenses will have a lower Strehl ratio degradation than large-diameter aspherical lenses (*Figure 5*).

The impact of surface irregularity on the Strehl ratio also depends on the f/# of the lens. As a general rule, faster aspherical lenses, or aspherical lenses with smaller f/#, have a greater sensitivity to the impact of surface irregularity on the Strehl ratio. As an example, the *Figure 6* compares an f/2 lens with an f/0.75 lens (both with a diameter of 25 mm).

The above examples illustrate that the underlying structure of the surface irregularity can have a significant effect on the Strehl ratio of a lens, particularly at high spatial frequencies. The **slope tolerance** is a simple and effective way of limiting this effect. For a given PV irregularity limit, higher slopes are associated with higher spatial frequencies on the surface. So, when the PV irregularity of a surface and its slope are limited, the admissible number of periods is reduced (see Figure 7).

To evaluate spatial frequencies more directly, we can use a specification called power spectral density (PSD). This function is calculated by analyzing the Fourier transform of the surface irregularity map, which gives a two-dimensional plot of the surface in terms of spatial frequency components. Placing tolerances on this plot will thus directly limit the number of periods.

## Conclusion

Aspherical lenses are an extremely powerful tool for improving the performance of optical systems while reducing the number of elements and, consequently, size and weight. Whether in medical equipment, microscopes, smartphones or autonomous vehicles, they are increasingly important in all optics-based sectors.

It's important to appreciate the complexity of lens manufacture. An aspherical surface cannot be manufactured in the same way as a spherical surface. A series of undercut grinding and polishing techniques must be used to create a variable curvature. These methods create additional problems that need to be controlled in order to maximize the performance of an aspherical lens.

Irregularity is always an important parameter, as any deviation from the ideal shape will lead to increased transmitted wavefront error and reduced performance. There are, however, secondary effects to take into account, notably the average spatial frequency of the surface irregularity profile. A surface with higher frequencies will have reduced performance compared to an identical surface with lower frequencies. This effect is more pronounced for larger lenses and lenses with a smaller f/#. For this reason, it's important to consider the shape of the surface irregularity across the entire lens aperture to understand the real impact that surface irregularity will have on performance. Tools such as the power spectral density function and the irregularity slope value provide a useful way of limiting spatial frequency effects and really pushing performance to higher levels of accuracy.

When specifying complex optical components such as high-quality aspherical lenses, many other factors need to be taken into account in addition to those described in this article. Final results often depend on choosing the right manufacturing partner, with the right experience, tools and metrology to succeed.

#### References

- Strehl, Karl W. A. "Theory of the telescope due to the diffraction of light", Leipzig, 1894.
- Mahajan, Virendra N. "Strehl ratio for primary aberrations in terms of their aberration variance". JOSA 73.6 (1983): 860-861.
- Kasunic, Keith J.,
*Laser Systems Engineering*SPIE Press, 2016. (ISBN 9781510604278) - Lawson, Janice K., et al. "Specification of optical components using the power spectral density function." Optical Manufacturing and Testing. Vol. 2536. International Society for Optics and Photonics, 1995.
- Messelink, Wilhelmus A., et al, "Mid-spatial frequency errors of mass-produced aspheres", Proc. SPIE 10829, Fifth European Seminar on Precision Optics Manufacturing, August 7, 2018, DOI: 10.1117/12.2318663.