# Aspheric Surfaces - Part 1: Introduction to Aspherical Surfaces in Optical Design

In the world of optical design, aspherical surfaces have revolutionized the approach to lens and system optimization. Unlike traditional spherical surfaces, aspheres can significantly reduce aberrations and improve image quality, making them invaluable in modern optical systems. This two-part article series delves into the transformative impact of aspherical surfaces, beginning with their application in singlet and doublet lenses, and advancing to a detailed performance analysis of the double Gauss lens system using Zemax OpticStudio. The second part will focus on the Schmidt-Cassegrain telescope, showcasing how aspherical surfaces can optimize its design for superior compactness and minimized aberrations.

Authored By Akhil Dutt Vijayakumar

## Introduction to Aspheric Surfaces

Traditional spherical lenses, characterized by a constant radius of curvature, suffer from spherical aberration. This occurs because they focus on-axis and off-axis light rays to different points, degrading image quality. Spherical lenses are also susceptible to aberrations like coma (resulting in a comet-shaped spot diagram) and astigmatism (causing an elongated spot diagram along the horizontal axis). Aspheric lenses address these limitations. Their surface profile is more complex, with a radius of curvature that varies from the optical axis. This allows them to correct various optical aberrations, resulting in sharper and clearer images. Figure shown here illustrates the typical difference between spherical and aspheric lens profiles.

In Zemax, traditional lenses and mirrors are designed using standard surface definitions. These definitions rely on a single radius of curvature to determine the element's shape, which is also axially symmetric. In contrast, aspheric surfaces possess a more complex form and are defined using polynomial equations. These equations typically consist of two parts: the first term represents the standard spherical surface, while the remaining terms account for the aspheric deviations from that ideal shape.

### Even Asphere

z(r) is the sag (surface height) at a radial distance r from the optical axis.

k is the conic constant.

Ai​ are the aspheric coefficients for higher-order terms.

Aspheric surfaces are categorized into different types based on the variations within their sag equations. The most prevalent type, employed in the majority of designs, is the even asphere. Even aspheres utilize only even powers for the aspheric coefficients within the sag equation. Conversely, odd aspheres incorporate odd powers in their coefficients, used in specific scenarios. Zemax also offers extended aspheres and Q-type aspheres for further design flexibility. Additionally, varying the conic constant (k) allows for the creation of diverse surface profiles. However, in this study, we focus on the properties of aspheric surfaces themselves, so we assume a k value of zero.

### Extended Asphere

Extended aspheres offer a more versatile and comprehensive approach to defining aspheric surfaces. They incorporate both even and odd-order polynomial terms within their sag equation. This enables the creation of significantly more complex surface profiles compared to even aspheres. These complex shapes have the potential to correct a wider range of higher-order aberrations, ultimately leading to improved optical performance. Unlike even aspheres, which are typically limited to terms up to the 16th order, extended aspheres can expand up to a staggering 480th order, providing a much richer set of coefficients to tailor the surface for optimal performance.

### Q-type Asphere

In Zemax OpticStudio, Qcon (Q-type Conic) and Qbfs (Q-type Best-Fit Sphere) are two distinct types of aspheric surface definitions.

Qcon(Q-type Conic)

The Qcon surface, also referred to as "Type 1" in Zemax, defines the sag departure of an aspheric surface from a base conic shape. It is particularly useful for cases where there are strong aspheric deviations from a sphere. The Qcon surface is characterized by its ability to represent the sag departure quantitatively, making it easier to judge the contribution of each term to the surface shape. The orthogonal terms of Qcon allow for straightforward optimization as they minimize interaction between the coefficients, which improves manufacturability and metrology processes​.

Qbfs (Q-type Best-Fit Sphere)

On the other hand, the Qbfs surface, often referred to as "Type 0", represents the sag departure from a best-fit sphere. This type is generally used when the surface closely resembles a sphere, and the deviations are not as pronounced as those typically addressed by the Qcon surface. The terms in Qbfs also benefit from orthonormality, which simplifies the optimization process by minimizing the cross-interaction between coefficients. This makes it easier to achieve precise control over the surface's slope and curvature during optimization​.

## Aspheric Design Exploration

Here in this study, we explore the powerful characteristics of aspheres in reducing the aberration for common optical systems such as singlet lens, doublet lens and double gauss system.

## Singlet

A singlet lens with an aperture size of 20mm is designed in Zemax and it is optimised to its minimum spot size. This is relatively small optical system in which the lens curvature, thickness is varied and material substitution is done to get the smallest spot as shown in the figure. The optimised system has a spot size of 23.052 µm and it is not diffraction limited. As this is a simple system it can be further optimised by varying the conic constant.

Varying conic constant

The system can optimise further by setting the conic constant of the lens as a variable and to run the same optimisation algorithm. The optimization wizard chooses a meniscus lens and the spot size is reduced further to a diffraction limited one.

Since the system is already optimized below diffraction limited value using conic sections, incorporating an aspheric surface may not be necessary for further improvement.

## Doublet

A doublet system with an aperture size of 15mm and a full field of view of 2.5o is designed in Zemax. Here we added 3 field points in the design to study the off-axis aberration effects. The system is optimised using the DLS optimiser and also with the Hammer to get the minimum spot size for the starting design.

In the optimised design the on-axis field point is diffraction limited but the off-axis field points suffer from coma aberrations. To address this limitation, we can utilize the aspheric tools in Zemax for further optimization. Specifically, the "Find Best Asphere" tool (located under the Optimize menu) helps identify the ideal surface for conversion into an asphere. This approach aims to minimize the overall merit function, ultimately resulting in a design with improved off-axis performance.

We retain the original configuration for comparison and analyze the modified design incorporating the aspheric surface. In many cases, the "Find Best Asphere" tool identifies even aspheres as the optimal solution.

Modified design

The design has been further optimized by incorporating an even asphere. We will now analyze the performance variations compared to the previous design.

Here an 8th order even asphere is added in the design, resulting in an improvement in the spot size of off-axis field points. The table shown here clearly demonstrates this improvement, along with minor reductions in aberrations.

## Double Gauss System

The double Gauss system, a widely used design in camera models, is presented here in its optimized form. This design serves as a starting point for exploring the impact of different aspheric surfaces when replacing standard elements.

The double Gauss system, a symmetric design optimized to provide better aberration control, exhibits excellent performance on-axis. However, the spot diagram reveals that while the on-axis field point is aberration-free, spherical aberration tends to increase for field points greater than 9.8 degrees. To address this limitation, we will replace the standard surface with different types of aspheres and compare their performance. Additionally, the sag profiles of these aspheres will be compared to understand the trade-off between design complexity and manufacturability. The RMS spot vs. field plot for the starting design is shown here.

Comparison of different Aspheres

We employed the 'Find Best Asphere' tool to identify the optimal surface for aspherization. To maintain the design's symmetry, we limited the variable values to the radius of curvature while keeping the thickness fixed. The tool identified the second surface as the optimal candidate for conversion into an even asphere. The LDE is shown below

The new spot diagram shows a reduction in spot size for on-axis and near-axis field points with small angles. Spherical aberration also decreases for these points but has a minimal effect on off-axis points with large angles.

Having observed the impact of asphericity on the second surface, we can now explore replacing it with different asphere types. This will allow us to compare their effects on spot size variation and analyze the resulting sag profiles.

RMS spot size Vs field comparison

In this design, we're replacing a standard surface with an aspheric one. The "best asphere" tool identified the second surface as the optimal location for this change. Having evaluated the impact of an even asphere in the previous step, we can now explore other asphere types for this surface. The second surface is then replaced with the other three aspheric types: extended asphere, Q-type asphere (Type 1 and Type 2). Five polynomials were used to define each asphere during the optimization process. The following figure shows the variation in spot size at a wavelength of 486nm for these different asphere types. The combined graph clearly demonstrates an improvement in spherical aberration and a reduction in spot size when introducing an asphere into the system. Interestingly, there is no significant difference in performance between extended and Q-type aspheres. However, all asphere types fail to reduce spherical aberration for off-axis field points at higher angles.

When introducing two aspheres in the system results in a much better solution that can be seen as red color in the graph and its design details will be discussed in the coming section.

The following figures show the surface sag cross sections, with the base radius removed, for even asphere, extended asphere, and Q-type aspheres (both Qbfs and Qcon) named as a,b,c,d in order respectively.

Each asphere exhibits a distinct sag profile, with all featuring a somewhat parabolic shape. However, the Qbfs asphere has a unique profile that presents additional manufacturing challenges. Since both the extended asphere and Q-type asphere were optimized using the same number of polynomials, their manufacturing complexity is likely comparable.

Double gauss system with multiple aspheres

The double gauss system has four lenses in which we can try some other combination with more aspheres to get a better solution. The best asphere tool sometime fail to provide the ideal solution if we are using multiple aspheres.

Here in this design we have replaced the two surfaces of the lens 4 with an even asphere with a 10th order polynomial. The spot diagram for this configuration shows a much significant reduction in its value than that of a single aspheric solution.

The RMS vs. field plot indicates that for a wavelength of 486nm, field points below 7 degrees are diffraction limited. Additionally, the spot size remains smaller for other wavelengths as well.

Field Curvature and Distortion plot comparison

The incorporation of an aspheric lens in the optimized system has yielded remarkable improvements in image quality. Here we are making a comparison of the distortion plot of the double gauss system having no aspheres and the system with 2 aspheres. The distortion plot demonstrates a dramatic reduction in distortion, with a decrease from 0.507% to a mere 0.062%. This shift likely represents a change from barrel distortion to a lesser degree of pincushion distortion. Furthermore, the field curvature plot reveals a significant flattening effect. The initial system exhibited sagittal and tangential curvatures of 0.1807mm and 0.1020mm, respectively. In the optimized design, these values have improved to 0.1351mm and 0.1188mm, indicating a flatter image plane and sharper focus across the field of view. These results strongly suggest that the aspheric lens has played a critical role in correcting aberrations and achieving a superior optical performance.

## Conclusion

This article has discussed singlet, doublet, and double Gauss systems, exploring how aspheres can be used to correct their aberrations. We've also explored different aspheric types used in Zemax and their associated sag profiles. Leveraging this understanding, the next part of this series will delve into the Schmidt-Cassegrain telescope, examining how aspheres play a crucial role in reducing its tube length and correcting aberrations.

### References:

1. All About Aspheric Lenses | Edmund Optics
2. Distortion | Edmund Optics
3. Singlet, Doublet and Double Gauss System can be found in the Sample file of the Zemax Root Folder. If Zemax is installed on the default directory this is:
• ...\Documents\Zemax\Samples\Sequential\Objectives