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Revisiting a classic lens design problem
Journal Pre-proof Revisiting a Classic Lens Design Problem Furkan E. Sahin
PII:
S0030-4026(20)30069-3
DOI:
https://doi.org/10.1016/j.ijleo.2020.164235
Reference:
IJLEO 164235
To appear in:
Optik
Received Date:
8 December 2019
Revised Date:
15 January 2020
Accepted Date:
16 January 2020
Please cite this article as: Furkan E. Sahin, Revisiting a Classic Lens Design Problem, (2020), doi: https://doi.org/10.1016/j.ijleo.2020.164235
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier.
Elsevier Editorial System(tm) for Optik – International Journal for Light and Electron Optics Manuscript Draft Manuscript Number: IJLEO-D-19-04728R2 Title: Revisiting a Classic Lens Design Problem Article Type: Full length article Section/Category: Optical Design Keywords: Optical Design; Optimization; Algorithms; Lens Design; Aberrations; Lens Design Software
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Corresponding Author: Dr. Furkan E Sahin, Corresponding Author's Institution: Maxim Integrated First Author: Furkan E Sahin Order of Authors: Furkan E Sahin
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Abstract: With global and local lens design optimization algorithms running on modern high-speed multi-core computers, highly optimized lens designs can be achieved quickly, even when started from scratch. Six years after two lens design experts compared a human designer's lens to what a global optimization algorithm can achieve, it is now time to revisit this classic lens design problem to demonstrate the level of performance that can be achieved with global optimization running on a state-of-the-art computer and no human designer intervention. Even though high-speed computers can iterate over many systems very quickly and converge on an optimized result, a human designer is still crucial in order to design the optimization approach and program the appropriate merit functions.
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Response to Reviewers: The reference style now matches the style of Optik journal.
Author Agreement
Author Agreement
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No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication.
*Impact Statement
Impact Statement Lens design and optimization is an important area of optical engineering, with applications in design of cameras, microscopes, telescopes and other optical instruments. Success in lens design is typically highly correlated with the starting point lens design, designer’s level of experience, and also familiarity with the optical design package being used. Traditionally, lens designers run optimization on desktop computers and workstations. Global optimization is typically a resource-heavy process, where many lenses are generated and evaluated to arrive at a near optimal design. Due to this resource-heavy nature, global optimization runs are typically performed over multiple days.
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Optical design software packages include proprietary lens optimization algorithms for local and global optimization. In this paper, a custom global optimization algorithm is run without user intervention and optimal results are achieved in just one hour.
*Cover Letter
Dear Prof. Tschudi, Please find attached my manuscript "Revisiting a Classic Lens Design Problem", which I would like to submit for publication as an original research article in the journal Optik. The manuscript summarizes the results of my research on global optimization of a classical lens design problem. This work is original and is not under review for publication at any other journal.
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Lens design and optimization is an important area of optical engineering, with applications in design of cameras, microscopes, telescopes and other optical instruments. Success in lens design is typically highly correlated with the starting point lens design, designer’s level of experience, and also familiarity with the optical design package being used.
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In this paper, a custom global optimization algorithm is run without user intervention and optimal results are achieved in just one hour.
The corresponding author is
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Dr. Furkan E. Sahin
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I am looking forward to hearing from you. Sincerely,
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Furkan E. Sahin
*Detailed Response to Reviewers
Revisiting a Classic Lens Design Problem Response to Reviewers’ Comments
I would like to thank the editorial team and the reviewer for their time and constructive comments. The recommendations for improving the manuscript have been addressed, and the paper is in publishable condition now.
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1. Well reported manuscript. Minor suggested review comments may improve the quality of the work. * Thank you for the kind words and your time to review the paper.
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2. Add more keywords * More and relevant keywords have been added to the paper.
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3. The motivation, contribution and novelty needs to be clearly mentioned in the Introduction section. * In order to clarify, the introduction has been edited. * There are now clear sentences: i. The motivation of this paper is to revisit this lens design problem and evaluate what level of performance optimization can be achieved with a contemporary optical design suite running on a modern desktop computer. ii. The novel contribution of the paper is pre-programming of the optimization approach and then accomplishing results with no human intervention in a time-limited optimization run. 4. Add a separate literature review section. * A new section (2. Developments in Complex Lens Design) has been added to the manuscript to include an in-depth literature review and provide example application areas.
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5. More analysis and discussion needs to be added. * A separate discussion section has been added. 6. It will be great if the authors add more case studies and application area. * Application area has been added and explained in the newly-added section 2. * In this particular paper, I wanted to compare to a reference design that is known to be very good. It is typically hard to find such good reference designs, that is why the paper is based on this particular case study. 7. Add last access date beside the url in the reference section. * “Last accessed” date has been added next to URLs in the reference section. Missing URLs for several references have been included.
*Manuscript
Revisiting a Classic Lens Design Problem Furkan E. Sahin Maxim Integrated, San Jose, CA, 95134, USA
Abstract
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With global and local lens design optimization algorithms running on modern high-speed multi-core computers, highly optimized lens designs can be achieved quickly, even when started from scratch. Six years after two lens design experts compared a human designer’s lens to what a global optimization algorithm can achieve, it is now time to revisit this classic lens design problem to demonstrate the level of performance that can be achieved with global optimization running on a state-of-the-art computer and no human designer intervention. Even though high-speed computers can iterate over many systems very quickly and converge on an optimized result, a human designer is still crucial in order to design the optimization approach and program the appropriate merit functions. Keywords: Optical Design, Optimization, Algorithms, Lens Design, Aberrations, Lens Design Software
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Lens design is a multi-parameter constrained optimization problem. Lens element parameters and spacings are perturbed to arrive at a final design that meets system constraints and desired performance targets. Achieving minimal degradation with as-built lenses considering system fabrication tolerances [1–3], stray light mitigation [4] and designing for robust operation over multiple operating conditions [5] further complicate the design and optimization process. Due to the computational complexity of ray-tracing and optimization algorithms, computers have been widely used for lens design since their earliest days [6, 7]. With high-speed modern computers and novel optimization algorithms [8–12], good lenses can be designed quickly. However, in-depth knowledge and prior experience of the human designer is still critical for starting from a good lens design, guiding the optimization process and arriving at very good lenses. Two very experienced optical designers worked on a challenge of comparing results from state-of-the-art optical design software [13] and a human designer (Man vs. Machine: a Lens Design Challenge [14]). They published the results in a 2013 proceedings paper [14], and one of the conclusions was that the human designer was more effective in coming up with an optimized design solution to a very hard lens design problem and in limited amount of time. Six years later, we now have faster computers with and multi-core processors. The motivation of this paper is to revisit this lens design problem and evaluate what level of performance optimization can be achieved with a contemporary optical design suite running on a modern desktop
computer. To this end, the same lens design problem was started from scratch, optimized with a two-stage global optimization approach and without human intervention. Promising results were achieved after only one hour of optimization. The novel contribution of the paper is pre-programming of the optimization approach and then accomplishing results with no human intervention in a time-limited optimization run.
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1. Introduction
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Email address: [email protected] (Furkan E. Sahin) Preprint submitted to Optik
2. Developments in Complex Lens Design Lens design is an important sub-field of optics with numerous consumer [15, 16], medical [17, 18], defense [19, 20] and industrial [21–24] applications. As the application areas of imaging systems and displays require higher performance typically with more complex structures, novel lens design approaches and methodologies have become active areas of research and development both in academia and in relevant industries. Miniature lens assemblies with injection-molded plastic lens elements is commonly found in modern smartphones. High quality imaging in smartphones is enabled by new lens design forms with highly aspheric surface profiles [3, 25, 26], high-precision manufacturing to fabricate and assemble these lenses [27, 28], new approaches for testing and validation [29] and system-level design with computational photography [15]. The great potential of augmented and virtual reality (AR/VR) displays for consumer applications has attracted significant interest from top companies including Google, Facebook and Microsoft [30]. In order to achieve unobtrusive, robust, reliable and manufacturable wearable displays, significant progress in freeform lenses has been made January 15, 2020
[16, 31, 32] and many university and industry research teams are actively working in this area to further advance the technology. 3. Lens Design Problem The lens design problem proposed by David Shafer is an interesting and challenging one [14]. The goal is to design a well corrected (diffraction-limited) and fast (low f-number) lens system that can operate at two widely separated laser lines, and achieve this with the fewest number of lens elements. Due to the polychromatic system requirement, correction of the chromatic variation in spherical aberration, coma and astigmatism is essential.
Specification 8 glass elements 100 mm > 10 mm < 250 mm 0.35 10 mm × 10 mm image 351 nm and 446.1 nm BK7 and LF5 <0.5% optical distortion < 0.5◦
Figure 1: Modulation transfer function (MTF) plot for David Shafer’s 8-element lens design. The lens drawing is shown in the inset figure. (From: [14]).
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Elements Effective Focal Length Back Focal Length System Length Numerical Aperture Field of View Wavelengths Lens Materials Distortion Chief Ray Angle
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Parameter
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Table 1: Parameters and specifications of the lens design problem.
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The parameters and specifications for the lens design problem are summarized in Table 1. In addition, the lens system should be free of vignetting. There was no mention of center and edge thickness constraints on the elements. In order to prevent very thin and very thick elements, element edge thicknesses were constrained to be greater than 1 mm and center thicknesses were constrained to be between 1 mm and 50 mm for the designs presented in this paper. There is also no mention of aspherical surfaces, therefore the design presented in this paper is limited to only spherical surfaces. In the referenced paper, authors started with an 11element lens design and aimed to reduce the number of elements without much compromise to the performance. The authors concluded that a diffraction-limited design can be achieved with only 8 elements, however they admit that this is indeed a very hard lens design problem for both human designers and computer algorithms. The location of the negative flint elements is important to achieve chromatic correction making the lens design problem challenging. Modulation transfer function plot and the lens drawing for the human designer (David Shafer) is shown in Figure 1. For this design, all three negative lenses are LF5 light flint glass. Don Dilworth was the designer working on the ”machine” side of the challenge. He is known for developing SYNOPSYS optical design software [13] which includes
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Figure 2: Modulation transfer function (MTF) plot for 8-element lens design solution achieved by Don Dilworth using SYNOPSYS optical design program [13] with no human designer intervention. The lens drawing is shown in the inset figure. (From: [14]).
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”Optimization Wizard” in OpticStudio was used. In addition to the default operands added by the wizard, several user operands were entered to constrain the effective focal length, maximum total track length, maximum distortion, maximum chief ray angle at the image plane and the minimum back focal distance. After the initial lens file was constructed (as shown in Figure 3), total optimization time was limited to about 1 hour. The first 30 minutes was dedicated to coming up with a good design with some level of aberration correction. This was achieved through Hammer Optimization (using Damped Least Squares algorithm) with root-meansquare (RMS) spot size merit function. Gaussian quadrature pupil sampling with 3 rings and 6 arms was selected in generating the merit function in order to have fast but also accurate modeling. Designer experience drove this decision that the initial stage of optimization should be to minimize the RMS spot size, as this would result in stable convergence from a poor initial design. With this merit function setup running on the particular hardware, OpticStudio iterated over about 85-million lens designs in 30 minutes and optimization was stopped automatically (triggered by the time elapsed) by the custom developed MATLAB script. The script then changed the merit function and the system was further optimized for another 30 minutes to improve the MTF. The starting point lens design consisted of a convexplano (with R = 98 mm) element, 6 flat plates and a plano-convex element. The radius of curvature of the last surface was defined as an f-number solve to meet the numerical aperture requirement, and this resulted in an initial radius of −98 mm. All elements were BK7 and the center thickness of each element was set at 5 mm, with 1 mm spacing in between elements. The aperture stop was placed as an external floating element at the front. The starting point lens design is shown in Figure 3. When the best location for the aperture stop needs to be identified, one method is to have a floating aperture stop [38]. In this case, the aperture stop is defined as the first surface of the system and the thickness of this surface is allowed to be negative during design optimization. Once optimization is concluded, the final location sets the aperture stop. This approach was taken to determine the optimal stop location for the lens design problem. The resulting lens design after the first 30 minutes of optimization is shown in Figure 4. In this case, the aperture stop is external to the lens system, about 55 mm from the vertex of the first lens element. Third, fifth and sixth elements from the left are LF5 and rest of the elements are BK7 glass. The modulation transfer function for this preliminary design is shown in Figure 5. After arriving at the preliminary solution, the design was further optimized (again with Hammer Optimization) to improve the MTF. For this purpose, the merit function was changed to a peak-to-valley (PV) wavefront error merit function. Denser pupil sampling, with an 8 × 8 rectangular array was selected. From designer experience,
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some of the latest developments in automatic lens design including the PSD III algorithm, DSEARCH and saddlepoint method [10, 33–35]. The 8-element design achieved with SYNOPSYS software with no human designer intervention was optimized in more than 8 hours and is shown in Figure 2. The approach presented in this paper is to use a proprietary global optimization function found in a commercial optical design software, namely Hammer Optimization in Zemax OpticStudio [36]. The goal is to arrive at an optimal solution with no human designer intervention. In order to eliminate human intervention, a MATLAB script was developed to control the optimization operation of OpticStudio. Beginning from scratch, initially a good design starting point was achieved and later this design was further refined with a different optimization merit function. As another example, CODE V optical design software can also be controlled by an external MATLAB script through the built-in API [37]. A similar approach can be taken with CODE V, and designers can probably arrive at similar results with the CODE V’s built-in global optimization engine.
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4. Lens Design Optimization
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As the goal of the paper is to reach an optimal solution from scratch, a global optimization algorithm needs to be employed. The Hammer Optimization algorithm in Zemax OpticStudio starts with a ray-traceable system and searches the solution space exhaustively for the optimum solution. When used together with the ”Glass Substitution” function in OpticStudio, Hammer Optimization can alter the lens materials to achieve chromatic correction. Following the requirements of the problem, ”Glass Substitution” was directed to use only BK7 and LF5 in the glass selection process.
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Figure 3: Lens design starting point for optimization. First and last surfaces are curved and other surfaces are planar.
Design optimization ran on a desktop computer with R CoreTM i7-9700K) and an 8-core, 3.6 GHz CPU (Intel 32 GB of RAM. When generating the merit functions, 3
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Figure 4: Lens design after 30 minutes of optimization with RMS spot size merit function. The aperture stop is external, about 55 mm in front of the front vertex of the first lens.
Figure 7: Modulation transfer function (MTF) for the lens design shown in Figure 6.
minimizing wavefront error helps with improving MTF, but the design starting point should have some level of performance [39]. Once again, the aperture stop location was defined as floating. Due to the computational complexity of the new merit function setup, time per iteration was longer this time and OpticStudio iterated over about 40-million lens designs in 30 minutes. The final lens design (after the second 30 minutes of optimization) is shown in Figure 6. The optimal location for the floating aperture stop was determined to be very close to the front surface of the second lens element by the optimization cycles. Therefore, the aperture stop was manually placed on this surface once the optimization was finalized. Third and sixth elements from the left are LF5 and rest of the elements are BK7 glass. The modulation transfer function for this design is shown in Figure 7. As seen from the MTF plot, this is a very good design with diffraction-limited performance across the field of view.
5. Discussion
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Figure 5: Modulation transfer function (MTF) for the lens design shown in Figure 4.
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The lens design in this paper started from scratch, that is only two surfaces are curved out of the 16 surfaces. This approach of starting with mostly flat plates worked in this particular case as the field of view was rather limited. This approach might fail when ray failures occur (when a ray either misses a surface or there is total internal reflection). If there is a ray failure during ray-tracing, optical design software is not able to calculate the merit function value. Therefore, it is important that the initial lens design is raytraceable. However, CODE V and OpticStudio can escape from ray failure cases when they occur during optimization [40]. Ray failures can be problematic with wide-angle lenses or with low f-number lenses, in which large curvatures are typical.
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6. Conclusions Modern lens design relies heavily on computer modeling, simulation and optimization. With contemporary 4
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Figure 6: Final lens design achieved after one hour of optimization.
multi-core computers and improved global optimization algorithms, it is easier and quicker to come up with good design forms that meet system criteria and performance goals. Six years after two experienced lens designers completed the man vs. machine challenge for lens design, the same problem was evaluated once again. It was interesting to see that starting from scratch, a very good lens design can be achieved after just one hour of optimization with no human intervention along the way. However, it was the knowledgeable lens designer who generated the appropriate merit functions and programmed a multi-stage optimization approach. High speed computation allows an experienced designer to come up with highly optimized designs quickly and efficiently.
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Acknowledgments
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[3] F. E. Sahin, Lens design for active alignment of mobile phone cameras, Optical Engineering 56 (6) (2017) 065102. [4] S. Grabarnik, Optical design method for minimization of ghost stray light intensity, Applied optics 54 (10) (2015) 3083–3089. [5] F. E. Sahin, Long-range, high-resolution camera optical design for assisted and autonomous driving, Photonics 6 (2) (2019) 73. doi:10.3390/photonics6020073. [6] D. P. Feder, Optical calculations with automatic computing machinery, J. Opt. Soc. Am. 41 (9) (1951) 630–635. doi:10.1364/JOSA.41.000630. [7] C. Wynne, Lens designing by electronic digital computer: I, Proceedings of the Physical Society 73 (5) (1959) 777. [8] F. E. Sahin, Open-source optimization algorithms for optical design, Optik 178 (2019) 1016–1022. [9] T. Houllier, T. L´ epine, Comparing optimization algorithms for conventional and freeform optical design: erratum, Optics Express 27 (20) (2019) 28383–28383. [10] Z. Hou, I. Livshits, F. Bociort, Practical use of saddle-point construction in lens design, in: Optical Design and Engineering VII, Vol. 10690, International Society for Optics and Photonics, 2018, p. 1069007. [11] C. Menke, Application of particle swarm optimization to the automatic design of optical systems, in: Optical Design and Engineering VII, Vol. 10690, International Society for Optics and Photonics, 2018, p. 106901A. [12] F. Sahin, A. Tanguay, Distortion optimization for wide-angle computational cameras, Optics express 26 (5) (2018) 5478–5487. [13] Dilworth, Donald C, The SYNOPSYS lens design program, https://www.osdoptics.com/ Last Accessed: January 13, 2020. [14] D. C. Dilworth, D. Shafer, Man versus machine: a lens design challenge, in: Current Developments in Lens Design and Optical Engineering XIV, Vol. 8841, International Society for Optics and Photonics, 2013, p. 88410G. [15] F. E. Sahin, R. Laroia, Light l16 computational camera, in: Applied Industrial Optics: Spectroscopy, Imaging and Metrology, Optical Society of America, 2017, pp. JTu5A–20. [16] O. Cakmakci, J. P. Rolland, Examples of hwd architectures: Low-, mid-and wide-field of view designs, Handbook of Visual Display Technology (2012) 2195–2211. [17] R. J. Duffey, R. W. Zabel, R. L. Lindstrom, Multifocal intraocular lenses, Journal of Cataract & Refractive Surgery 16 (4) (1990) 423–429. [18] F. Sahin, B. McIntosh, P. Nasiatka, J. Weiland, M. Humayun, A. Tanguay, Eye-tracked extraocular camera for retinal prostheses, in: Frontiers in Optics, Optical Society of America, 2015,
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I would like to thank SPIE for the permission to reproduce figures from the paper: D. C. Dilworth and D. Shafer, Man versus machine: a lens design challenge, in Current Developments in Lens Design and Optical Engineering XIV, , vol. 8841 (International Society for Optics and Photonics, 2013), vol. 8841, p. 88410G.
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Disclosures
The author declares no conflicts of interest. References [1] D. C. Dilworth, New tools for the lens designer, in: Current Developments in Lens Design and Optical Engineering IX, Vol. 7060, International Society for Optics and Photonics, 2008, p. 70600B. [2] J. P. McGuire, Designing easily manufactured lenses using a global method, in: International Optical Design Conference, Optical Society of America, 2006, p. TuA6.
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*Declaration of Interest Statement
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
PII:
S0030-4026(20)30069-3
DOI:
https://doi.org/10.1016/j.ijleo.2020.164235
Reference:
IJLEO 164235
To appear in:
Optik
Received Date:
8 December 2019
Revised Date:
15 January 2020
Accepted Date:
16 January 2020
Please cite this article as: Furkan E. Sahin, Revisiting a Classic Lens Design Problem, (2020), doi: https://doi.org/10.1016/j.ijleo.2020.164235
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier.
Elsevier Editorial System(tm) for Optik – International Journal for Light and Electron Optics Manuscript Draft Manuscript Number: IJLEO-D-19-04728R2 Title: Revisiting a Classic Lens Design Problem Article Type: Full length article Section/Category: Optical Design Keywords: Optical Design; Optimization; Algorithms; Lens Design; Aberrations; Lens Design Software
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Corresponding Author: Dr. Furkan E Sahin, Corresponding Author's Institution: Maxim Integrated First Author: Furkan E Sahin Order of Authors: Furkan E Sahin
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Abstract: With global and local lens design optimization algorithms running on modern high-speed multi-core computers, highly optimized lens designs can be achieved quickly, even when started from scratch. Six years after two lens design experts compared a human designer's lens to what a global optimization algorithm can achieve, it is now time to revisit this classic lens design problem to demonstrate the level of performance that can be achieved with global optimization running on a state-of-the-art computer and no human designer intervention. Even though high-speed computers can iterate over many systems very quickly and converge on an optimized result, a human designer is still crucial in order to design the optimization approach and program the appropriate merit functions.
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Response to Reviewers: The reference style now matches the style of Optik journal.
Author Agreement
Author Agreement
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No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication.
*Impact Statement
Impact Statement Lens design and optimization is an important area of optical engineering, with applications in design of cameras, microscopes, telescopes and other optical instruments. Success in lens design is typically highly correlated with the starting point lens design, designer’s level of experience, and also familiarity with the optical design package being used. Traditionally, lens designers run optimization on desktop computers and workstations. Global optimization is typically a resource-heavy process, where many lenses are generated and evaluated to arrive at a near optimal design. Due to this resource-heavy nature, global optimization runs are typically performed over multiple days.
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Optical design software packages include proprietary lens optimization algorithms for local and global optimization. In this paper, a custom global optimization algorithm is run without user intervention and optimal results are achieved in just one hour.
*Cover Letter
Dear Prof. Tschudi, Please find attached my manuscript "Revisiting a Classic Lens Design Problem", which I would like to submit for publication as an original research article in the journal Optik. The manuscript summarizes the results of my research on global optimization of a classical lens design problem. This work is original and is not under review for publication at any other journal.
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Lens design and optimization is an important area of optical engineering, with applications in design of cameras, microscopes, telescopes and other optical instruments. Success in lens design is typically highly correlated with the starting point lens design, designer’s level of experience, and also familiarity with the optical design package being used.
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In this paper, a custom global optimization algorithm is run without user intervention and optimal results are achieved in just one hour.
The corresponding author is
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Dr. Furkan E. Sahin
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I am looking forward to hearing from you. Sincerely,
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Furkan E. Sahin
*Detailed Response to Reviewers
Revisiting a Classic Lens Design Problem Response to Reviewers’ Comments
I would like to thank the editorial team and the reviewer for their time and constructive comments. The recommendations for improving the manuscript have been addressed, and the paper is in publishable condition now.
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1. Well reported manuscript. Minor suggested review comments may improve the quality of the work. * Thank you for the kind words and your time to review the paper.
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2. Add more keywords * More and relevant keywords have been added to the paper.
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3. The motivation, contribution and novelty needs to be clearly mentioned in the Introduction section. * In order to clarify, the introduction has been edited. * There are now clear sentences: i. The motivation of this paper is to revisit this lens design problem and evaluate what level of performance optimization can be achieved with a contemporary optical design suite running on a modern desktop computer. ii. The novel contribution of the paper is pre-programming of the optimization approach and then accomplishing results with no human intervention in a time-limited optimization run. 4. Add a separate literature review section. * A new section (2. Developments in Complex Lens Design) has been added to the manuscript to include an in-depth literature review and provide example application areas.
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5. More analysis and discussion needs to be added. * A separate discussion section has been added. 6. It will be great if the authors add more case studies and application area. * Application area has been added and explained in the newly-added section 2. * In this particular paper, I wanted to compare to a reference design that is known to be very good. It is typically hard to find such good reference designs, that is why the paper is based on this particular case study. 7. Add last access date beside the url in the reference section. * “Last accessed” date has been added next to URLs in the reference section. Missing URLs for several references have been included.
*Manuscript
Revisiting a Classic Lens Design Problem Furkan E. Sahin Maxim Integrated, San Jose, CA, 95134, USA
Abstract
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With global and local lens design optimization algorithms running on modern high-speed multi-core computers, highly optimized lens designs can be achieved quickly, even when started from scratch. Six years after two lens design experts compared a human designer’s lens to what a global optimization algorithm can achieve, it is now time to revisit this classic lens design problem to demonstrate the level of performance that can be achieved with global optimization running on a state-of-the-art computer and no human designer intervention. Even though high-speed computers can iterate over many systems very quickly and converge on an optimized result, a human designer is still crucial in order to design the optimization approach and program the appropriate merit functions. Keywords: Optical Design, Optimization, Algorithms, Lens Design, Aberrations, Lens Design Software
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Lens design is a multi-parameter constrained optimization problem. Lens element parameters and spacings are perturbed to arrive at a final design that meets system constraints and desired performance targets. Achieving minimal degradation with as-built lenses considering system fabrication tolerances [1–3], stray light mitigation [4] and designing for robust operation over multiple operating conditions [5] further complicate the design and optimization process. Due to the computational complexity of ray-tracing and optimization algorithms, computers have been widely used for lens design since their earliest days [6, 7]. With high-speed modern computers and novel optimization algorithms [8–12], good lenses can be designed quickly. However, in-depth knowledge and prior experience of the human designer is still critical for starting from a good lens design, guiding the optimization process and arriving at very good lenses. Two very experienced optical designers worked on a challenge of comparing results from state-of-the-art optical design software [13] and a human designer (Man vs. Machine: a Lens Design Challenge [14]). They published the results in a 2013 proceedings paper [14], and one of the conclusions was that the human designer was more effective in coming up with an optimized design solution to a very hard lens design problem and in limited amount of time. Six years later, we now have faster computers with and multi-core processors. The motivation of this paper is to revisit this lens design problem and evaluate what level of performance optimization can be achieved with a contemporary optical design suite running on a modern desktop
computer. To this end, the same lens design problem was started from scratch, optimized with a two-stage global optimization approach and without human intervention. Promising results were achieved after only one hour of optimization. The novel contribution of the paper is pre-programming of the optimization approach and then accomplishing results with no human intervention in a time-limited optimization run.
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1. Introduction
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Email address: [email protected] (Furkan E. Sahin) Preprint submitted to Optik
2. Developments in Complex Lens Design Lens design is an important sub-field of optics with numerous consumer [15, 16], medical [17, 18], defense [19, 20] and industrial [21–24] applications. As the application areas of imaging systems and displays require higher performance typically with more complex structures, novel lens design approaches and methodologies have become active areas of research and development both in academia and in relevant industries. Miniature lens assemblies with injection-molded plastic lens elements is commonly found in modern smartphones. High quality imaging in smartphones is enabled by new lens design forms with highly aspheric surface profiles [3, 25, 26], high-precision manufacturing to fabricate and assemble these lenses [27, 28], new approaches for testing and validation [29] and system-level design with computational photography [15]. The great potential of augmented and virtual reality (AR/VR) displays for consumer applications has attracted significant interest from top companies including Google, Facebook and Microsoft [30]. In order to achieve unobtrusive, robust, reliable and manufacturable wearable displays, significant progress in freeform lenses has been made January 15, 2020
[16, 31, 32] and many university and industry research teams are actively working in this area to further advance the technology. 3. Lens Design Problem The lens design problem proposed by David Shafer is an interesting and challenging one [14]. The goal is to design a well corrected (diffraction-limited) and fast (low f-number) lens system that can operate at two widely separated laser lines, and achieve this with the fewest number of lens elements. Due to the polychromatic system requirement, correction of the chromatic variation in spherical aberration, coma and astigmatism is essential.
Specification 8 glass elements 100 mm > 10 mm < 250 mm 0.35 10 mm × 10 mm image 351 nm and 446.1 nm BK7 and LF5 <0.5% optical distortion < 0.5◦
Figure 1: Modulation transfer function (MTF) plot for David Shafer’s 8-element lens design. The lens drawing is shown in the inset figure. (From: [14]).
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Elements Effective Focal Length Back Focal Length System Length Numerical Aperture Field of View Wavelengths Lens Materials Distortion Chief Ray Angle
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Table 1: Parameters and specifications of the lens design problem.
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The parameters and specifications for the lens design problem are summarized in Table 1. In addition, the lens system should be free of vignetting. There was no mention of center and edge thickness constraints on the elements. In order to prevent very thin and very thick elements, element edge thicknesses were constrained to be greater than 1 mm and center thicknesses were constrained to be between 1 mm and 50 mm for the designs presented in this paper. There is also no mention of aspherical surfaces, therefore the design presented in this paper is limited to only spherical surfaces. In the referenced paper, authors started with an 11element lens design and aimed to reduce the number of elements without much compromise to the performance. The authors concluded that a diffraction-limited design can be achieved with only 8 elements, however they admit that this is indeed a very hard lens design problem for both human designers and computer algorithms. The location of the negative flint elements is important to achieve chromatic correction making the lens design problem challenging. Modulation transfer function plot and the lens drawing for the human designer (David Shafer) is shown in Figure 1. For this design, all three negative lenses are LF5 light flint glass. Don Dilworth was the designer working on the ”machine” side of the challenge. He is known for developing SYNOPSYS optical design software [13] which includes
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Figure 2: Modulation transfer function (MTF) plot for 8-element lens design solution achieved by Don Dilworth using SYNOPSYS optical design program [13] with no human designer intervention. The lens drawing is shown in the inset figure. (From: [14]).
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”Optimization Wizard” in OpticStudio was used. In addition to the default operands added by the wizard, several user operands were entered to constrain the effective focal length, maximum total track length, maximum distortion, maximum chief ray angle at the image plane and the minimum back focal distance. After the initial lens file was constructed (as shown in Figure 3), total optimization time was limited to about 1 hour. The first 30 minutes was dedicated to coming up with a good design with some level of aberration correction. This was achieved through Hammer Optimization (using Damped Least Squares algorithm) with root-meansquare (RMS) spot size merit function. Gaussian quadrature pupil sampling with 3 rings and 6 arms was selected in generating the merit function in order to have fast but also accurate modeling. Designer experience drove this decision that the initial stage of optimization should be to minimize the RMS spot size, as this would result in stable convergence from a poor initial design. With this merit function setup running on the particular hardware, OpticStudio iterated over about 85-million lens designs in 30 minutes and optimization was stopped automatically (triggered by the time elapsed) by the custom developed MATLAB script. The script then changed the merit function and the system was further optimized for another 30 minutes to improve the MTF. The starting point lens design consisted of a convexplano (with R = 98 mm) element, 6 flat plates and a plano-convex element. The radius of curvature of the last surface was defined as an f-number solve to meet the numerical aperture requirement, and this resulted in an initial radius of −98 mm. All elements were BK7 and the center thickness of each element was set at 5 mm, with 1 mm spacing in between elements. The aperture stop was placed as an external floating element at the front. The starting point lens design is shown in Figure 3. When the best location for the aperture stop needs to be identified, one method is to have a floating aperture stop [38]. In this case, the aperture stop is defined as the first surface of the system and the thickness of this surface is allowed to be negative during design optimization. Once optimization is concluded, the final location sets the aperture stop. This approach was taken to determine the optimal stop location for the lens design problem. The resulting lens design after the first 30 minutes of optimization is shown in Figure 4. In this case, the aperture stop is external to the lens system, about 55 mm from the vertex of the first lens element. Third, fifth and sixth elements from the left are LF5 and rest of the elements are BK7 glass. The modulation transfer function for this preliminary design is shown in Figure 5. After arriving at the preliminary solution, the design was further optimized (again with Hammer Optimization) to improve the MTF. For this purpose, the merit function was changed to a peak-to-valley (PV) wavefront error merit function. Denser pupil sampling, with an 8 × 8 rectangular array was selected. From designer experience,
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some of the latest developments in automatic lens design including the PSD III algorithm, DSEARCH and saddlepoint method [10, 33–35]. The 8-element design achieved with SYNOPSYS software with no human designer intervention was optimized in more than 8 hours and is shown in Figure 2. The approach presented in this paper is to use a proprietary global optimization function found in a commercial optical design software, namely Hammer Optimization in Zemax OpticStudio [36]. The goal is to arrive at an optimal solution with no human designer intervention. In order to eliminate human intervention, a MATLAB script was developed to control the optimization operation of OpticStudio. Beginning from scratch, initially a good design starting point was achieved and later this design was further refined with a different optimization merit function. As another example, CODE V optical design software can also be controlled by an external MATLAB script through the built-in API [37]. A similar approach can be taken with CODE V, and designers can probably arrive at similar results with the CODE V’s built-in global optimization engine.
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4. Lens Design Optimization
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As the goal of the paper is to reach an optimal solution from scratch, a global optimization algorithm needs to be employed. The Hammer Optimization algorithm in Zemax OpticStudio starts with a ray-traceable system and searches the solution space exhaustively for the optimum solution. When used together with the ”Glass Substitution” function in OpticStudio, Hammer Optimization can alter the lens materials to achieve chromatic correction. Following the requirements of the problem, ”Glass Substitution” was directed to use only BK7 and LF5 in the glass selection process.
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Figure 3: Lens design starting point for optimization. First and last surfaces are curved and other surfaces are planar.
Design optimization ran on a desktop computer with R CoreTM i7-9700K) and an 8-core, 3.6 GHz CPU (Intel 32 GB of RAM. When generating the merit functions, 3
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Figure 4: Lens design after 30 minutes of optimization with RMS spot size merit function. The aperture stop is external, about 55 mm in front of the front vertex of the first lens.
Figure 7: Modulation transfer function (MTF) for the lens design shown in Figure 6.
minimizing wavefront error helps with improving MTF, but the design starting point should have some level of performance [39]. Once again, the aperture stop location was defined as floating. Due to the computational complexity of the new merit function setup, time per iteration was longer this time and OpticStudio iterated over about 40-million lens designs in 30 minutes. The final lens design (after the second 30 minutes of optimization) is shown in Figure 6. The optimal location for the floating aperture stop was determined to be very close to the front surface of the second lens element by the optimization cycles. Therefore, the aperture stop was manually placed on this surface once the optimization was finalized. Third and sixth elements from the left are LF5 and rest of the elements are BK7 glass. The modulation transfer function for this design is shown in Figure 7. As seen from the MTF plot, this is a very good design with diffraction-limited performance across the field of view.
5. Discussion
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Figure 5: Modulation transfer function (MTF) for the lens design shown in Figure 4.
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The lens design in this paper started from scratch, that is only two surfaces are curved out of the 16 surfaces. This approach of starting with mostly flat plates worked in this particular case as the field of view was rather limited. This approach might fail when ray failures occur (when a ray either misses a surface or there is total internal reflection). If there is a ray failure during ray-tracing, optical design software is not able to calculate the merit function value. Therefore, it is important that the initial lens design is raytraceable. However, CODE V and OpticStudio can escape from ray failure cases when they occur during optimization [40]. Ray failures can be problematic with wide-angle lenses or with low f-number lenses, in which large curvatures are typical.
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6. Conclusions Modern lens design relies heavily on computer modeling, simulation and optimization. With contemporary 4
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Figure 6: Final lens design achieved after one hour of optimization.
multi-core computers and improved global optimization algorithms, it is easier and quicker to come up with good design forms that meet system criteria and performance goals. Six years after two experienced lens designers completed the man vs. machine challenge for lens design, the same problem was evaluated once again. It was interesting to see that starting from scratch, a very good lens design can be achieved after just one hour of optimization with no human intervention along the way. However, it was the knowledgeable lens designer who generated the appropriate merit functions and programmed a multi-stage optimization approach. High speed computation allows an experienced designer to come up with highly optimized designs quickly and efficiently.
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Acknowledgments
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[3] F. E. Sahin, Lens design for active alignment of mobile phone cameras, Optical Engineering 56 (6) (2017) 065102. [4] S. Grabarnik, Optical design method for minimization of ghost stray light intensity, Applied optics 54 (10) (2015) 3083–3089. [5] F. E. Sahin, Long-range, high-resolution camera optical design for assisted and autonomous driving, Photonics 6 (2) (2019) 73. doi:10.3390/photonics6020073. [6] D. P. Feder, Optical calculations with automatic computing machinery, J. Opt. Soc. Am. 41 (9) (1951) 630–635. doi:10.1364/JOSA.41.000630. [7] C. Wynne, Lens designing by electronic digital computer: I, Proceedings of the Physical Society 73 (5) (1959) 777. [8] F. E. Sahin, Open-source optimization algorithms for optical design, Optik 178 (2019) 1016–1022. [9] T. Houllier, T. L´ epine, Comparing optimization algorithms for conventional and freeform optical design: erratum, Optics Express 27 (20) (2019) 28383–28383. [10] Z. Hou, I. Livshits, F. Bociort, Practical use of saddle-point construction in lens design, in: Optical Design and Engineering VII, Vol. 10690, International Society for Optics and Photonics, 2018, p. 1069007. [11] C. Menke, Application of particle swarm optimization to the automatic design of optical systems, in: Optical Design and Engineering VII, Vol. 10690, International Society for Optics and Photonics, 2018, p. 106901A. [12] F. Sahin, A. Tanguay, Distortion optimization for wide-angle computational cameras, Optics express 26 (5) (2018) 5478–5487. [13] Dilworth, Donald C, The SYNOPSYS lens design program, https://www.osdoptics.com/ Last Accessed: January 13, 2020. [14] D. C. Dilworth, D. Shafer, Man versus machine: a lens design challenge, in: Current Developments in Lens Design and Optical Engineering XIV, Vol. 8841, International Society for Optics and Photonics, 2013, p. 88410G. [15] F. E. Sahin, R. Laroia, Light l16 computational camera, in: Applied Industrial Optics: Spectroscopy, Imaging and Metrology, Optical Society of America, 2017, pp. JTu5A–20. [16] O. Cakmakci, J. P. Rolland, Examples of hwd architectures: Low-, mid-and wide-field of view designs, Handbook of Visual Display Technology (2012) 2195–2211. [17] R. J. Duffey, R. W. Zabel, R. L. Lindstrom, Multifocal intraocular lenses, Journal of Cataract & Refractive Surgery 16 (4) (1990) 423–429. [18] F. Sahin, B. McIntosh, P. Nasiatka, J. Weiland, M. Humayun, A. Tanguay, Eye-tracked extraocular camera for retinal prostheses, in: Frontiers in Optics, Optical Society of America, 2015,
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I would like to thank SPIE for the permission to reproduce figures from the paper: D. C. Dilworth and D. Shafer, Man versus machine: a lens design challenge, in Current Developments in Lens Design and Optical Engineering XIV, , vol. 8841 (International Society for Optics and Photonics, 2013), vol. 8841, p. 88410G.
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Disclosures
The author declares no conflicts of interest. References [1] D. C. Dilworth, New tools for the lens designer, in: Current Developments in Lens Design and Optical Engineering IX, Vol. 7060, International Society for Optics and Photonics, 2008, p. 70600B. [2] J. P. McGuire, Designing easily manufactured lenses using a global method, in: International Optical Design Conference, Optical Society of America, 2006, p. TuA6.
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*Declaration of Interest Statement
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: