## Abstract

Mid-spatial-frequency (MSF) surface errors are left after processing aspheres or freeforms by small bits. MSF errors sometimes have significant effects on the image quality of optical systems. The efforts on the analysis and tolerancing of the MSF errors have been paid. This paper focuses on the desensitization to the MSF errors. The design target for the desensitization to the MSF errors is defined. A possibility of the desensitization is shown for optical systems in which freeforms are used with spheres.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The optical system is very sensitive to the manufacturing errors. The typical manufacturing errors are the surface shape error, the thickness error, and the decenter. To achieve the best as-built performance, the desensitization is very important. The author developed the general method of the desensitization to the manufacturing errors [1]. The effect of the manufacturing errors is evaluated with the raytrace data of the nominal system and controlled as the optimization targets. The thickness error or the decenter are simple as variables. But the surface shape error is more complicated. If the surface shape error can be expressed as the combination of the basic patterns, the desensitization can be done to these basic patterns [2–4].

Mid-spatial-frequency (MSF) surface errors are left after processing aspheres or freeforms by small bits. MSF errors sometimes have the significant effects on the image quality of optical systems. As the title of the paper of Forbes indicates, there are various challenges in the treatment of the MSF errors. Forbes listed the challenges related to the MSF errors as (i) define and quantify that structure, (ii) measure it, (iii) estimate its optical impact, and (iv) characterize the spectral capabilities and signatures of the production processes in use [5]. Youngworth estimated its impact on the RMS OPD and observed that its impact depends on three things as (i) the variance of the MSF errors, (ii) the difference in refractive indices on either side of the surfaces, and (iii) the wavelength [6]. Tamkin investigated the impact of the structured MSF errors to the image quality [7].

If the optical system is less sensitive to the MSF errors, the performance of the optical system would be better under the same level of the MSF errors. But no one has mentioned the desensitization to the MSF errors, at my best knowledge. The purpose of this paper is to investigate the possibility of the desensitization to the MSF errors. The impact of the MSF errors to the image quality such as the optical transfer function has been investigated. But these methods have no direct relation to the ordinary optimization of optical systems. The purpose of this paper is to define the optimization target which is easy to implement in the ordinary optimization.

The pattern of the MSF errors are random and has the large degree of freedom. It is impossible to clearly define these patterns and control the sensitivity of each component. In this paper the effect of the MSF errors is estimated with the loss of the RMS OPD. The structured MSF errors may cause the diffraction and ghost images [8]. The desensitization to random MSF errors is expected to be effective to structured MSF errors as well.

In section 2, the optimization target for the desensitization is defined. In section 3, a design example is explained. In section 4, the conclusion is given.

## 2. Definition of the optimization target for the desensitization

There are discussions about the definition of the MSF errors [9]. In this paper, the suppositions in the following derivation should be considered to be the definition of the MSF errors. The wave aberration with the manufacturing errors is written as

where ${W_0}(\vec{u})$ is the nominal wave aberration, ${\Gamma _i}(\vec{u})$ is Gamma factor on the*i*

^{th}surface, ${d_i}(\vec{u})$ is the deviation of the surface from the nominal position to the direction of the surface normal on the

*i*

^{th}surface, and $\vec{u}$ is the pupil coordinate [10]. ${\Gamma _i}(\vec{u})$ and ${d_i}(\vec{u})$ are the functions of the pupil coordinate because the pupil coordinate determines the cross point of the ray with each surface. The wave aberration is the function of the field position, but the field coordinate is suppressed for the simplicity. Gamma factor is defined as where $\vec{n}$ is the unit vector normal to the surface at the point of the refraction, $\vec{l}$ is the direction vector of the incident ray, $\vec{l}^{\prime}$ is the direction vector of the refracted ray, and

*N*and

*N*’ are the refractive indices before and after the surface. Figure 1 shows the quantities to calculate the Gamma factor at the surface 1.

If the position of the surface changes by *d* to the direction of the surface normal, the optical path length changes by $\Gamma d$. The surface deviation *d* has the specific pattern depending on the types of manufacturing errors such as shape error, thickness error, or decenter. Gamma factor is calculated with the raytrace data of the sample rays for the aberration control. With values of $\Gamma d$ over the pupil, the loss of the RMS OPD by each manufacturing error can be estimated and controlled during the optimization. Only the raytrace on the nominal system is necessary to estimate the sensitivity of the various manufacturing errors. Figure 2 shows the Gamma factor as a function of the sine of incident angle for *N *= 1, *N*'=1.5 and *N *= 1, *N*'=1.8. The value is mainly determined by the relative refractive index rather than the incident angle.

It is supposed that the manufacturing errors can be separated to the slow variation part and the fast variation part to the pupil coordinate. This is justified because the MSF errors are left from the processing with a small bit. The slow variation part includes the surface shape error, the thickness error, and the decenter. The fast variation part includes the MSF errors and the manufacturing errors of the higher spatial frequencies.

To calculate the RMS OPD, the wave aberration and the square of the wave aberration are integrated. The integral of the fast variation part can be supposed to vanish.*i*

^{th}surface. The desensitization to the MSF errors is to reduce the average of $\Gamma _i^2$ over the pupil. Practically, the average of $\Gamma _i^2$ over the sample rays for the aberration control is calculated at each surface.

## 3. Design example

#### 3.1 Specifications

The design example is the anamorphic fisheye lens. When the fisheye lens is used to see the wide field, it may happen that the wide field is not necessary to the vertical direction. The purpose of the anamorphic fisheye lens is to improve the resolution by narrowing the field of the vertical direction. The axially non-symmetric components are necessary for the anamorphic optical system. In this example, the optical system composed of spheres and freeforms is used. The total lens count of spheres and the total lens count of freeforms are supposed. Table 1 shows the outline of specifications. The anamorphic ratio is 3/2.

#### 3.2 Design process

The design consists of 2 steps. The first step is the design with 10 spherical lenses. The full field of view is 180deg. The starting point is the parallel plates. The global optimization with escape function is applied [11]. In the second step, 3 spherical lenses are replaced to freeform lenses. To determine the position of the freeform lenses, the method of the traveling freeform is applied [12]. The shape of the freeform is expressed with the 2-dim. Qcon [13,14]. The order of the polynomials is up to the 8th order. The freeform is symmetric to the horizontal axis and the vertical axis respectively. The glass material is optimized with the fictitious glass model.

The designs without and with the sensitivity control to the MSF errors of the freeforms are compared. The degree of the MSF errors at each freeform is supposed to be similar. The design target is the sum of the average of $\Gamma _i^2$ of freeforms. The target value is 60% of the value without the sensitivity control.

#### 3.3 Result

Figure 3(a) is the design without the sensitivity control. Figure 3(b) is the design with the sensitivity control. The upper drawings are the horizontal section. The lower drawings are the vertical section. The lenses drawn in red are freeform lenses. The refractive index at d-line (Nd) and Abbe number at d-line (Vd) of each glass is shown. For example 516.641 means Nd = 1.516 and Vd = 64.1. Nd.Vd of the negative lenses is shown on the upper side. Nd.Vd of the positive lenses is shown on the lower side.

Figure 4(a) is the MTF field plot of the design without the sensitivity control. Figure 4(b) is the MTF field plot of the design with the sensitivity control. The circles show the MTF values at 20 positions on the image plane. The center of the circle shows the position on the image plane. The lower left is the center of the image plane. The upper right is the corner of the image plane. Two circles are drawn on each position. The black circle shows the diffraction limited MTF. The color circle shows the MTF with the aberrations. The length of the line from the center of the circle shows the MTF value to the direction of the line. The spatial frequency is 160lp/mm. This chart shows the full information of MTF of the axially non-symmetric optical system.

Figure 5(a) shows the total sensitivity to MSF errors. Figure 5(b) shows the sensitivity of each freeform at the center. The number written on the horizontal axis is the surface number. The stop surface is not counted. The reduction of the sensitivity is achieved mainly with the change of the position of the second freeform lens. From Fig. 3 and Fig. 5(b), the reduction of the sensitivity is achieved mainly with the lower refractive indices of the freeform lenses. The effect of the MSF errors is smaller if the refractive index is lower.

Figure 6(a) shows the asymmetry of the base quadric of the freeforms. The value is the difference between the curvatures of horizontal and vertical directions at the center. Figure 6(b) shows the deviation from the base quadric of the freeforms. The value is the integral of the square of the deviation over the surface. With the sensitivity control, the shape of freeforms did not become more eccentric.

## 4. Conclusion

In this paper, it was shown that the average of square of Gamma factor on the pupil expresses the sensitivity to the MSF surface errors. It was also shown that the desensitization to the MSF errors is possible for the optical system in which aspheres or freeforms are used with spheres. The best configuration was achieved automatically with the method of the traveling freeform and the fictitious glass model. The pattern of the MSF errors is various. But the proposed design target is expected to be effective to any types of MSF errors.

## Disclosures

The author declares no conflicts of interest.

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