**Key specifications and their impact on performance**

## Introduction

Any spherical optical surface, even those designed and manufactured perfectly, will exhibit spherical aberration. Due to this inherent defect in spherical surfaces, incident light rays focus at different points during the formation of an image and create a blur. It is to correct this defect that aspherical lenses were designed. (Figure 1).

Aspherical lenses can offer an improved spot size, smaller by several orders of magnitude than spherical lenses, which almost completely eliminates blur and significantly improves image quality. Aspherical lens elements also allow 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 affecting 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 rotational symmetry lenses whose radius of curvature varies from the center to the edge. This geometry poses unique challenges that are not found in traditional lens manufacturing. 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 the same time. In contrast, the continuously variable radius of curvature of an aspherical lens requires underopening polishing by tools small enough to create different curvatures located at different locations on the surface.

The design and manufacture of aspherical lenses are both fundamentally more complex than those of spherical components; so it's important to know their individual specifications and what that 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 vertex. 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 conical constant. The terms A**

*k*_{4,}A

_{6}and A

_{8}are called aspherical coefficients of

^{4th,}

^{6th}and

^{8th}order. Figure 2 illustrates the comparison between an aspherical surface and a spherical surface.

The aspherical surface described by the above equation represents the ideal shape; the goal of manufacturing is to get as close as possible. Inevitably, there will be a deviation from the ideal surface profile: this is called the asphere 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 referred to as a peak-to-valley (P-V) value, which represents the difference between the maximum and minimum deviation points. However, this value can be misleading because 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 multi-point surface and calculates an average value for the entire optics. This value can vary from a few microns for commercial-grade optics to a few tenths of a micron for high-precision optics.

Although the surface irregularity gives a good indication of the performance of the lens, a significant amount of information is still missing. If we consider the entire optical surface, the deviation from the ideal shape at a given point is not constant. In addition, the grinding and underopening polishing techniques used in the manufacture of the aspherical lens can create repetitive patterns and structures *in* the surface irregularity profile, known as medium spatial frequencies. Another key specification that results from this is the irregularity slope or **slope tolerance.** This value sets an upper limit to the rate of change of the assphere figure error, describing how quickly the deviation from the ideal shape can change in a given window. Typical values range from 1 μm/mm for commercial grade to 0.15 μm/mm for high precision. The size of the window is an important part of the specification and should be chosen so as to be less than the wavelength of the targeted average 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 report is a specification for comparing the actual performance of an optical system to its performance limited by diffraction. For aspherical lenses and other focusing optics, the Strehl ratio is defined as the ratio between the maximum focal point irradiance of the manufactured optics and the maximum irradiance limited by diffraction^{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 wavefront error transmitted RMS using the following approximation, in which represents the WAVEFRONT error RMS in waves^{2}. This approximation is valid for transmitted wavefront error values < 0,1 ondes.

The Strehl ratio of an optics is highly dependent on the accuracy of its surface, 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 the surface irregularity is modeled as a cosine function with rotational symmetry, we can explore the resulting Strehl ratio as a function of the RMS surface irregularity for a variety of cosine periods*(Figure 3* and *Figure 4).*

Here, the key factor is not the period of the cosine in mm, but the *number of periods on the opening of the lens*. For a given underopening 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 the surface irregularity on the Strehl ratio also depends on the f/# of the lens. Typically, faster aspherical lenses, or aspherical lenses with smaller f/# lenses, have greater sensitivity to the impact of surface irregularity on the Strehl ratio. For example, Figure *6* compares an f/2 lens to 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, especially in the case of large spatial frequencies. Slope **tolerance** is a simple and effective way to limit this effect. For a given PV irregularity limit, higher slopes are associated with higher spatial frequencies on the surface. Thus, when the PV irregularity of a surface and its slope are limited, the permissible number of periods is reduced (see Figure 7).

To evaluate spatial frequencies more directly, a specification called power spectral density (DSP) can be used. This function is calculated by analyzing the Fourier transformation of the surface irregularity map that gives a two-dimensional plot of the surface in terms of spatial frequency components. The placement of tolerances on this route will thus directly limit the number of periods.

## Conclusion

Aspherical lenses are an extremely powerful tool to improve 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 sectors based on optics.

It is important to appreciate the complexity of lens manufacturing. An aspherical surface cannot be manufactured in the same way as a spherical surface. A series of grinding and under-opening 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 an increase in the transmitted wavefront error and a decrease in performance. There are, however, side effects to consider, including 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 large lenses and lenses with a smaller f/#. For this reason, it is important to consider the shape of the surface irregularity over the entire lens opening to understand the real impact that the surface irregularity will have on performance. Tools such as power spectral density function and irregularity slope value provide a useful way to limit spatial frequency effects and actually push performance to higher levels of accuracy.

When specifying complex optical components like high-quality aspherical lenses, there are many other factors to consider in addition to those described in this article. The end 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, 7 August 2018, DOI: 10.1117/12.2318663.