## Abstract

In the final stage of lens design, it is usually a critical step to balance the optical performance of a lens system across the sampled fields, which is achieved by adjusting the weights to these fields. Because the current optical design software packages use fixed weights in the optimization process, the task of weight adjustment is left to the optical designer, who has to change the weights manually after each optimization trail. However, this process may take a very long time to finish, especially when many fields of the lens system are sampled, and the results are subjectively affected by the designer’s design experience. In this paper, we propose an automatic performance balancing method. An automatic outer loop is added in the optimization process. The weight for each sampled field and azimuth is calculated appropriately according to the actual performance of the current design and the system requirements, and it is applied to the corresponding field and azimuth automatically in the next optimization trial. The method is successfully implemented in CODE V, and design examples show that it is very effective.

© 2010 OSA

## 1. Introduction

Aberration balancing is very important in lens design. It is also essential to balance the system performance across the entire field of view (FOV). The latter is achieved by adjusting the weights to the sampled fields and azimuths during the optimization process. This is at present a major task for optical designers because the current optical design software packages do not change the field weights in their optimization features. At the beginning of the optimization, one may use the default weights provided by the software, or a more proper set of weights need to be assigned, which are usually determined according to the designer’s experience. After each optimization trial, the designer needs to manually adjust the weights to the sampled fields and azimuths according to the design requirements and the current image quality of each field and azimuth, and the system is then reoptimized again with the original constraints but with the newly-assigned weights. It can be a long and tedious process to balance the image quality and maximize the total performance of the optical system, which may require many trials of this iterative optimization run. For a conventional single-configuration optical system with spherical surfaces and rotational symmetry, three to seven sampled fields in the tangential section are usually sufficient. For a zoom lens, the number of sampled fields is multiplied by the number of sampled configurations. For modern designs adopting aspheric surfaces and/or asymmetric structures [1,2], more fields have to be sampled to accurately describe the aberration characteristics and the system performance. For systems without symmetry, the sampling has to cover the entire FOV instead of being represented by a single (tangential) section. It is difficult for a designer to decide the weights for so many fields, which is one of the reasons why optical design is so time consuming and heavily dependent on the designer’s experience. If the weight for each sampled field can be automatically determined and assigned for each optimization trial, the balancing burden of the final stage of optical system design would be remarkably reduced and the effectiveness of the balancing would be improved.

An automatic image performance balancing method is proposed in this study. An outer loop is added in the optimization process, in which the weight for each field and azimuth is calculated appropriately and adjusted automatically before each optimization trial. It can usually achieve the performance balancing across the FOV with several automatic optimization trials, and it can often improve the overall performance of the system. The automatic weighting method and its implementation in CODE V [3] are discussed in Section 2, and the design examples are presented in Section 3.

## 2. Automatic image performance balance method

The error function in lens design is a number that describes the overall quality of a lens system, which is a function of the construction parameters. The smaller the value of this function, the closer the optical system is to the desired state. The goal of optimization is to reduce this error function as much as possible [4–7].

The error function *ω* is usually composed of weighted aberration terms, which can be expressed as [3]:

*Z*is the configuration indicator, and

*nz*is the total number of configurations;

*F*is the field indicator, and

*nf*is the total numbers of sampled fields; λ is the wavelength indicator, and

*nw*is the total number of wavelengths;

*R*is the ray indicator, and

*nr*is the total number of traced rays.

The set of weights are tailored to the designer’s needs since it does allow for cross terms. It uses field weight arrays (W_{X}, W_{Y}) for the X and Y components of the aberrations (Δ*x*, Δ*y*) as a function of field and zoom, wavelength weights (W_{W}) as a function of wavelength and zoom, and aperture weights (W_{A}) as a function of ray and zoom position.

The weights for the wavelength W_{W}(Z, λ) are determined by the spectral response of the optical system, for example, an optical designer often uses the weight 1, 2 and 1 for designing an eyepiece with the wavelength at 456, 589 and 656nm. The weights for the aperture in CODE V are defined as

*x*) are the ray coordinates in the pupil, A is the normalization factor for this weight, and the value of α is used to shift emphasis from the center of the pupil to the edge.

_{p}, y_{p}The only remaining issue is how to determine the optimal values for W_{X}(*Z*, *F*) and W_{Y}(*Z*, *F*). An easily-understood guideline can be used to calculate these weights, that is to set lower values to the fields and azimuths whose performances are close to the desired system requirements, and higher values to those whose performances are further away from the system requirements. For systems with a uniform performance requirement, the principle can be simply stated as: assign lower weights to the fields and azimuths with better performances and higher weights to those with worse performances, and the automatic adjustments of the field weights during the optimization process can be achieved by

*i*is the number of the next optimization iteration;

*m*and

_{h}*m*are scaling factors; W

_{l}_{X0}(

*Z*,

*F*) and W

_{Y0}(

*Z*,

*F*) are the initial weight values for the first iteration, which are all set to 100 in this study;

*ζ*

_{X}(

*Z*,

*F*) and

*ζ*

_{Y}(

*Z*,

*F*) specify the image performances for a specified zoom position and field point in the sagittal and tangential planes, respectively, such as the modulation transfer function (MTF) values for a specified spatial frequency, or the inverse of spot diagram diameter; and is the average performance across the entire FOV, defined as

For systems whose performance requirements are not uniform across the FOV, Eqs. (5) and (6) similar to (3) and (4) can be used for automatic image performance balancing by replacing the value of with the design targets ${\zeta}_{x}{}_{t}(Z,F)$ and ${\zeta}_{y}{}_{t}(Z,F)$ for different fields and azimuths. This will be shown in the first design example.

Note that the weights for different fields and azimuths are relative values. They can be normalized before each optimization trial, and the normalization functions can be defined as

Convergence is an important factor affecting the optimization process. The basic assumption made is that increasing the relative weight will improve the performance of the corresponding field and azimuth, while reducing the relative weight will likely deteriorate the performance. Based on this assumption, the differences between the average or target performance and the actual performance of each field and azimuth will become smaller and smaller after each optimization trial. Therefore, the change in the weight for each field and azimuth is to become smaller. This guarantees the convergence and the stabilization of the weights and the optimization process in most cases. However, two types of exit conditions are set up to circumvent the possible infinite loop problem. One condition is defined as the improvement value of each optimization trial, and the optimization will be terminated if the improvement is less than a specified value ε. The other condition is set by incorporating a counter for the total number of optimization trials, and the optimization will be terminated if the counter exceeds a specified number n.

Because the weight adjustment method is based on the image performance, a relatively good starting point is needed for the performance-balancing optimization, and a sufficiently large number of variables are required to make the assumptions mentioned above valid. Under some conditions such as lack of optimization variables or too many constraints, no matter what kind of optimization method is used, the performance of the system across the entire FOV cannot be well balanced and improved.

Figure 1(a) shows the process of the traditional manual image performance balancing, and Fig. 1(b) shows the flow chart of the proposed automatic balancing method, where an outer loop is added for the automatic weight adjustment for optimization. In this latter method, the initial weights for all the sampled fields and azimuths are set to an equal value of 100. The image performance is evaluated before each optimization trial, and the weights for the next trial are calculated accordingly. The sampled fields and azimuths with performance values lower than the average will be assigned higher weights, and vice versa. The weight will be multiplied by 100 during normalization. The exit condition is examined after each optimization trial. A macro is written in CODE V to realize the concept, which is used successfully in many test and practical designs, and by employing the macro, more balanced systems are obtained than without using the automatic method, not mentioning that a lot of time can be saved in the endeavor to balance the design. A few of the design examples are presented below.

## 3. Design examples

In this section, three examples, a projection system with rotational symmetry, a zoom system, and a freeform system without rotational symmetry, will be used to demonstrate the effectiveness of the automatic image performance balancing method described in the previous section. Among the three examples, the first one is adopted from Ref [8], and the starting point for our method had already been balanced manually. The starting points for the second and third examples were the results of local optimization in CODE V, in which the weights for all fields and azimuths were set to the same value. The local optimization for each of the designs had reached a local minimal and become stagnated. Previously, in order to further improve the design and balance the image performance, the designer would have to start the manual balancing process illustrated in Fig. 1(a) or apply global optimization, which is time consuming. In the automatic balancing method proposed in this paper, diffraction MTF values at specified spatial frequencies are selected as the evaluation criteria since the MTF curves often can present a comprehensive assessment of the performance in imaging systems. For all the examples, the value of α in Eq. (2) is set to be 0.5. The scaling factors, *m _{h}* and

*m*in Eqs. (3) through (6), are set to be 1 and 2, respectively. The maximum number of optimization trials for exit condition is set to 20 in order to study the behavior of the weight variance.

_{l}By using the automatic balancing method, a designer may not obtain exactly the same solution as the examples, depending on the weights applied and the starting point, but the solutions should have a similar performance. After a few trials, the designer can find a set of weights for both azimuths that will allow the lens to meet the MTF and other specifications through the automatic balancing method.

#### 3.1 Example 1 – projection lens with 5 sampled fields

The lens system of example 1 is used for a head-mounted projection display proposed by Hua and Rolland [8]. The exit pupil is 12 mm, and the EFL is 35mm. This design is rotationally symmetric and it has 5 sampled fields in the tangential section. The full diagonal FOV is 52.4°. The weights of the five fields were manually adjusted by Hua and Rolland during their optimization process, and the final weights were 1.0, 0.8, 0.8, 0.5, and 0.3 for each field, respectively. The weights for the wavelengths are 1, 2 and 1 for 656.3nm, 550.0nm and 456.1nm, respectively.

Figure 2(a) shows the layout of the system. The MTF of each field and azimuth evaluated at the Nyquist spatial frequency 33lps/mm are used to construct the weight function with the goal of pushing the entire FOV to have equally good performance. Figures 2(b) and 2(c) plot the polychromatic MTF curves for the full 12-mm pupil for the manually adjusted design [8] and for the result after applying our automatic balancing method, respectively. While most of the MTF curves in Fig. 2(b) are between 0.2 and 0.5 at 33lps/mm, all the curves in Fig. 2(c) are above 0.5 at 33lps/mm. Although the lens shapes remain similar to the original design, the performances of the newly balanced design become better balanced among different fields and azimuths, and the overall performance is also greatly improved. This is more clearly shown in Fig. 2(d), where the mean, RMS and spread values of MTF for the sampled fields and azimuths are plotted at eleven spatial frequencies equally spaced between 0 and 33lps/mm. The mean value is symbolized by a horizontal line. The range of spread of the MTF values for different fields and azimuths at a given spatial frequency is illustrated by connecting the highest and the lowest MTF values with a vertical line. The RMS is marked by a rectangle with the height as the RMS value and the vertical center at the mean. The MTF curves before and after the automatic balancing are shifted slightly to avoid overlapping of the vertical lines. The spread of the MTF curves among the sampled fields is greatly reduced from 0.35 to 0.12 at 33lps/mm. The average MTF spread value at these eleven spatial frequencies for the automatically balanced design is only one-third of the corresponding value for the manually balanced design.

To demonstrate the effectiveness of the automatic balance method, Fig. 3 shows the error function variation at different optimization trials of automatic balancing. The value dropped from 88 for the original design in [8] to 24.3 after 20 trials of optimization and the rapid descent occurred in the first 4 to 5 trials. Figure 4 further plots the average MTF value of the five sampled fields at 33lps/mm spatial frequency as a function of trial numbers. In agreement with the error function values, the average MTF across the sampled fields in the sagittal plane at 33lps/mm increased from 0.25 to 0.55, while the one in tangential direction from 0.35 to 0.52. The MTF difference between the two azimuths decreased from 0.1 to 0.03. The automatic balancing process was completed within one minute.

Figure 5 shows the weight variation curves versus the optimization trials. The curves become very stable after a few trials so that the convergence of further optimization is guaranteed. The final relative weights for the five fields are 0.37, 0.02, 0.05, 1.00 and 0.37 in the sagittal azimuth, and 0.51, 0.32, 0.15, 0.37 and 1.00 in the tangential azimuth. They are very different from the weights assigned by Hua and Rolland, and very difficult to arrive with the manual adjustment method.

In many cases, it is not necessary to push for the entire FOV to have the same performance. Frequently, it is preferable to have better performance in the center field than in the marginal fields. The average item $\overline{\zeta}$ in Eq. (3) and (4) can be replaced by the specified target, as defined in Eq. (5) and (6), to obtain the desired performance. To demonstrate the application of the automatic balancing method for achieving specified design goals, the same example as shown in Fig. 2(a) is utilized. Instead of balancing the sampled fields to have equal performance as shown in Fig. 2(c), this example requires the MTF targets at 33lps/mm to be 0.6 for the center and 0.2 fields, 0.5 for the 0.5 field, and 0.4 for the 0.7 and marginal fields. The same starting point as the previous example is adopted from [8].

Figure 6 shows the result optimized by the automatic balancing method with specified targets. The MTF curves were separated from each field and close to the design goal. The MTF value at 33lps/mm is 0.65 for the center field, 0.62 for the 0.2 field, 0.53 for the half field, and 0.42 for the 0.7 and marginal fields. As shown in Fig. 7 , the error function dropped very fast after the first trial, then became worse at the second and third trials, because the weights were over adjusted. However, the routine re-adjusted the weights automatically and pushed the design back toward convergence and achieved desired performance targets.

#### 3.2 Example 2 – zoom lens with 4 sampled fields and 3 configurations

The lens in Fig. 8 is adopted from US patent 3,464,763. It is a 9mm to 36mm zoom system with an F-number of 2.0. Four fields are sampled for 3 different configurations, with EFLs of 9mm, 20mm and 36mm, respectively. Therefore the lens has 12 sampled fields in total. All the radii of curvature, thicknesses and glasses are selected as optimization variables. The total lengths of three configurations are constrained to the same value. The minimal and maximum thicknesses of the lenses are also controlled during the optimization. The weights for all wavelengths are the same. The MTF of each field, azimuth and configuration evaluated at 200lps/mm are used to construct the weight function.

Most of the MTF values were above 0.2 at 200lps/mm after local optimization when weights for all fields and azimuths were kept same, as shown in Figs. 9(a), (c) and (e) . However, the MTF values are above 0.4 at 200lps/mm after the automatic weighting optimization method, as shown in Figs. 9(b), (d) and (f). The ratios between the average MTF spread at the ten spatial frequencies (equally spaced between 0 and 200lps/mm) before and after the automatic balancing are 1.63:1, 3.14:1 and 1.77:1 for the three sampled configuration, respectively.

The error function decreased from 2.3 to 1.6, as shown in Fig. 10 . The difference between the average MTF of all the sampled fields across the three configurations at the spatial frequency of 200lps/mm in the two azimuths decreased from 0.09 to 0.05, as shown in Fig. 11 . The performance is well balanced and the total performance is also improved. Figures 12(a) and (b) plotted the field weight variation curves versus the trials: most weights in sagittal direction turn to stable after a few trials and all weight curves in tangential direction become stable. This example required more trials for the weighting factors to converge than the previous example, mainly due to the increased complexity of the design. The balancing process was finished within five minutes.

#### 3.3 Example 3 – free-form lens with 15 sampled fields covering half of the entire FOV

The lens for this example is used in a head-mounted display system. The design consists of three freeform optical surfaces as shown in Fig. 13 , and 15 fields are used in optimization to balance the image quality over the FOV. The EFL is 19mm, the diagonal FOV is 33.4°, the exit pupil diameter is 7mm, and the eye clearance is 20mm. The description and design approaches can be found in Ref [9]. The MTF of each field and azimuth evaluated at 33lps/mm are used to build up the weight function. The weights for the wavelengths are 1, 2 and 1 for 656.3nm, 587.6nm and 486.1nm, respectively.

Figure 14 shows the error function variation versus the number of optimization trials. The value dropped dramatically from 213 to 74. As shown in Fig. 15, the average MTF across the sampled fields in the sagittal plane at 33lps/mm increased from 0.2 to 0.32, while the value in the tangential direction rises from 0.145 to 0.31. The average MTF difference between the two azimuths decreased from 0.045 to 0.01. It is obvious that significant improvement on overall image quality and performance balance is achieved with our method.

Figures 16 (a), (c), (e) show the MTF curves of the initial design after local optimization with equal weights for all the sample fields. Figures 16(b), (d), (f) show the MTF curves after applying the automatic performance balancing method. The mean and spread MTF at eleven spatial frequencies (equally spaced between 0 and 33lps/mm) are analyzed. The mean MTF of the sampled fields and azimuths at 33lps/mm is 0.18 and 0.32 before and after automatic balancing, respectively. The ratio between the average MTF spread at the eleven spatial frequencies before and after automatic balancing is 1.8:1.

Figure 17 and Fig. 18 show the MTF and the weight variation curves during the optimization, respectively. The MTFs converges to 0.3 gradually except for one field higher than the average value.

It is always necessary to explore global optimization in designing a freeform optical system [10]. Unfortunately, the optimization speed for such an optical system is terribly slow especially when several freeform surfaces are used. In this example, 3 freeform optical surfaces are included, over 120 parameters are defined as variables, and 15 fields are sampled. It costs several minutes to finish just a single local optimization loop and needs more than 250 hours to finish a global optimization loop. Further balancing is still needed even after the global optimization. However, by employing our automatic balancing method, the balancing process is finished within half an hour.

## 4. Conclusions

Finding the appropriate weights for the sampled fields and azimuths is very important in the final stage of lens optimization especially for complex lens systems. An image performance balancing method for different fields and azimuths is presented and implemented, the weights are automatically adjusted in an outer loop around the local optimization and the expected performances are achieved. Although the trial numbers were set to 20 for all the examples provided, five to ten trials are usually enough since the systems can be dramatically improved during the first several trials. The weights are changed at every step and the final distributions are so complex that it is not easy for a designer to determine manually. We would not try to compare the design results of our method with manually balanced results, as the latter really depends on the designer’s experience and the time that he or she can afford to spend on the final fine tune. The goal of this work is to present a method that can achieve image performance balancing automatically and effectively, without human interaction and dependence on the designer’s experience.

The results also show that correctly changing the weights for fields and azimuths in the error function help to overcome a stagnated optimization process, and a better design result can be achieved. This method will contribute to the improvement of the image performance across the entire FOV and the reduction of the final development time in lens design.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China (60827003), the Hi-Tech Research and Development Program of China (2009AA01Z308, 2006AA02Z4E5), the Innovation Team Development Program of the Chinese Ministry of Education (IRT0606). The work is also partially supported by a National Science Foundation Grant award 06-44446. We also thank Optical Research Associates for providing the educational license of the CODE V software.

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