Optical Design Problem Research Articles

Efficient evaluation of the Jacobian in the damped least-squares method for optical design problems using algorithmic differentiation.

The fast evaluation of the Jacobian is an essential part of the optimization of optical systems using the damped least-squares algorithm. While finite differences provide an intuitive way to approximate derivatives, algorithmic differentiation is a technique to evaluate them exactly. However, applying algorithmic differentiation to a ray tracing routine for optical systems with many parameters is computationally expensive, where the main costs are caused by the determination of the ray-surface intersection. To overcome this disadvantage, we present a mathematical analysis of the ray-surface intersection and its efficient differentiation in both forward and reverse mode algorithmic differentiation. Futhermore the structure of the optimization variables and operands is exploited to derive a method that allows computation of the Jacobian in the same order of computational complexity as the primal ray trace. The method is successfully tested for a rotationally symmetric lens system and a freeform design task.

Method of discontinuous optical glass optimization based on aberration correction in optical design.

Optical glass selection is an important research object in optical design, which is an important way to aberration correction. However, these methods to our knowledge either do not correct aberration well or consume too much time. To efficiently solve the apochromatic problems in optical design, this paper presents what we believe to be a novel automatic optimization method for discontinuous optical glass based on auto glass selection (AGS). The method first calculates several aberrations related to glass materials in the optical system to select candidate optical glasses, which include longitudinal aberration, lateral chromatic aberration, and Petzval sum. It then skillfully explores the candidate optical glass list for the optimization of discontinuous glass. Using a flat-field apochromatic microscope and a wide-angle lens as examples, this method demonstrates its superiority in aberration correction and efficiency, can be automated optimization, requires relatively less computational effort, and achieves fast and stable convergence of results.

Artificial intelligence in optical lens design

Traditional optical design entails arduous, iterative stages that significantly rely on the intuition and experience of lens designers. Starting-point design selection has always been the major hurdle for most optical design problem, and different designers might produce different final lens designs even if using the same initial specification. Lens designers typically choose designs from existing lens databases, analyse relevant lens structures, or explore patent literature and technical publications. With increased processing capability, producing automated lens designs using Artificial Intelligence (AI) approaches is becoming a viable alternative. Therefore, it is noteworthy that a comprehensive review addressing the latest advancements in using AI for starting-point design is still lacking. Herein, we highlight the gap at the confluence of applied AI and optical lens design, by presenting a comprehensive review of the current literature with an emphasis on using various AI approaches to generate starting-point designs for refractive optical systems, discuss the limitations, and suggest a potential alternate approach for further research.

Wavefront aberration determination in non-axially symmetrical optical systems.

Non-axially symmetrical optical systems can provide better solutions to several optical design problems. However, determining their wavefront aberrations is challenging due to the great diversity of their configurations. Accordingly, in the present study, the optical path length (OPL) between two boundaries in a non-axially symmetrical system is expanded by a Taylor series expansion with respect to the base ray. This expansion converts the OPL from a deep composite function to a polynomial form. The various order wavefront aberrations are then obtained from the coefficients of the polynomials in terms of the system independent variables. The proposed method is equally valid for axially symmetrical systems provided that the object is placed on the meridional plane. Overall, the method provides a systematic approach for analyzing the wavefront aberrations in a wide range of optical systems.

End-to-End Diverse Metasurface Design and Evaluation Using an Invertible Neural Network.

Employing deep learning models to design high-performance metasurfaces has garnered significant attention due to its potential benefits in terms of accuracy and efficiency. A deep learning-based metasurface design framework typically comprises a forward prediction path for predicting optical responses and a backward retrieval path for generating geometrical configurations. In the forward design path, a specific geometrical configuration corresponds to a unique optical response. However, in the inverse design path, a single performance metric can correspond to multiple potential designs. This one-to-many mapping poses a significant challenge for deep learning models and can potentially impede their performance. Although representing the inverse path as a probabilistic distribution is a widely adopted method for tackling this problem, accurately capturing the posterior distribution to encompass all potential solutions remains an ongoing challenge. Furthermore, in most pioneering works, the forward and backward paths are captured using separate models. However, the knowledge acquired from the forward path does not contribute to the training of the backward model. This separation of models adds complexity to the system and can hinder the overall efficiency and effectiveness of the design framework. Here, we utilized an invertible neural network (INN) to simultaneously model both the forward and inverse process. Unlike other frameworks, INN focuses on the forward process and implicitly captures a probabilistic model for the inverse process. Given a specific optical response, the INN enables the recovery of the complete posterior over the parameter space. This capability allows for the generation of novel designs that are not present in the training data. Through the integration of the INN with the angular spectrum method, we have developed an efficient and automated end-to-end metasurface design and evaluation framework. This novel approach eliminates the need for human intervention and significantly speeds up the design process. Utilizing this advanced framework, we have effectively designed high-efficiency metalenses and dual-polarization metasurface holograms. This approach extends beyond dielectric metasurface design, serving as a general method for modeling optical inverse design problems in diverse optical fields.

Transverse ray aberrations determination innon-axially symmetrical optical imaging systemsusing Taylor series expansion.

Non-axially symmetrical optical systems can provide better solutions to several optical design problems. However, the determination of their aberrations is a challenge due to the great diversity of their configurations. To address this problem, this study extends previous work by the present group to determine the transverse ray aberrations in non-axially symmetrical optical imaging systems through the use of a Taylor series expansion. The expansion converts the coordinates of the incident point, which are highly composite functions, of a general ray on the image plane into polynomial functions. The various order transverse ray aberrations are then extracted directly from the corresponding terms of the polynomials. Notably, the derived expressions are exact since they are determined without any approximations. Furthermore, the proposed method can be equally applied to axially symmetrical systems provided that the object is confined on the meridional plane. Consequently, the proposed method offers a useful and generally applicable approach for the systematic analysis of the transverse ray aberrations in optical systems with the object at either finite or infinite positions.

Different view on diffraction-limited imaging optics design.

The task of imaging optics design starts with choosing a subset of imaging parameters such as the object distance, the image magnification, the image resolution, and the depth of field. It is clear that the imaging parameters are inter-related and their appropriate trade-off is the key problem of final optical design. We provide a novel analysis of the optical design considering diffractive imaging phenomena and thin lens approximation, and we derive formulas for the solution of individual parameters. We show that combinations of three independent optical design input variables determine all other optical and geometrical parameters under diffraction-limited optical imaging. We also formulate an invariant relation in the object space if the depth of field is one of the design's target parameters.

Designing Freeform Optical Surfaces by the Monge-Ampère Equations: A Review

Background: The equations of Monge-Ampère type which arise in geometric optics are used to design illumination lenses and mirrors. The optical design problem can be formulated as an inverse problem: determine an optical system consisting of a reflector and/or refractor that converts a given light distribution of the source into a desired target light distribution. For two decades, the development of fast and reliable numerical design algorithms for the calculation of freeform surfaces for irradiance control in the geometrical optics limit is of great interest in current research. Objective: The objective of this paper is to summarize the types, algorithms, and applications of Monge-Ampère equations. It helps scholars better grasp the research status of Monge-Ampère equations and further explore the theory of Monge-Ampère equations. Methods: This paper reviews the theory and applications of Monge–Ampère equations from four aspects. We first discuss the concept and development of Monge–Ampère equations. Then we derive two different cases of Monge–Ampère equations. We also list the numerical methods of Monge–Ampère equation in actual scenes. Finally, the paper gives a brief summary and an expectation. Results: This paper reviews the theory and applications of Monge-Ampère equations from four aspects. We first discuss the concept and development of Monge-Ampère equations. Then we derive two different cases of Monge-Ampère equations. We also list the numerical methods of the Monge-Ampère equation in actual scenes. Finally, the paper gives a brief summary and an expectation. Conclusion: Monge-Ampère equation has been widely applied in the geometric optics field since the predetermined energy distribution and the boundary condition creation can be well satisfied. Although the freeform surfaces designed by the Monge-Ampère equations is developing rapidly, there is still plenty of room for development in the design of the algorithms.

Deep reinforcement learning with a critic-value-based branch tree for the inverse design of two-dimensional optical devices

In optical engineering, designing a device or a system with the desired property is an important but challenging task. It is relatively straightforward to compute the physical properties of a given design; however, there is no general method for the reverse, i.e., designing with desired properties. To address this problem with a computational method, this paper proposes a deep reinforcement learning-based inverse design framework consisting of two methods: Inverse DEsign Agent (IDEA) and Critic-Value-based Branch Tree (CVBT) algorithm. IDEA is a deep reinforcement learning model based on the Advantage Actor–Critic (A2C) method using a deep learning simulator as a replacement for a real environment, significantly reducing training time compared to conventional methods. The CVBT algorithm suggests several design candidates using the critic values of IDEA, while conventional methods propose only one candidate. In this study, the proposed framework was applied to a two-dimensional optical device design problem. Experimental results with untrained target properties demonstrated that the proposed model, IDEA-CVBT, achieved state-of-the-art performance in terms of accuracy and stability. For instance, in a scenario with a binary type design, IDEA-CVBT exhibited an accuracy of 91.5%, while conventional deep reinforcement models showed 36.1% and 7.4% accuracies. Extensive analyses verified that IDEA-CVBT can be employed as an assistance system for engineers since IDEA-CVBT suggests several appropriate candidates that satisfy target properties.

Bi-Directional Benes With Large Port-Counts and Low Waveguide Crossings for Optical Network-on-Chip

We propose a new variant of the Benes network using a merge-replace-fold approach. It is to realize a large port-count optical switch with low waveguide crossings. A bidirectional switch is built instead of a unidirectional switch. Compared to the classical Benes, it requires the same number of switches but far fewer intersections. The novel designs enable a 24-percentage reduction in waveguide intersections. Moreover, it has a worst-case insertion loss of 12.01 dB and a comparable crosstalk level of −13.67 dB between the neighbouring paths. Thus, it offers an energy-efficient way to solve the optical on-chip design problem in a bidirectional manner.

Inverse reflector design for a point source and far-field target

We present a method for the design of a single freeform reflector that converts the light distribution of a point source to a desired light distribution in the far field. Using the geometrical-optics law of reflection and requiring energy conservation, this optical design problem can be represented by a generalized Monge–Ampère equation for the shape of the reflector with transport boundary condition. We use a generalized least-squares algorithm that can handle a logarithmic cost function in the corresponding optimal transport problem. The algorithm first computes the optical map and subsequently constructs the optical surface. We demonstrate that the algorithm can generate reflector surfaces for a number of complicated target distributions.

Comparing optimization algorithms for conventional and freeform optical design.

Five search algorithms from the literature of black-box optimization were implemented and applied to optical design problems. These algorithms are the Particle Swarm Optimization, Gravity Search Algorithm, Cuckoo Search, Covariance Matrix Adaptation Evolution Strategy and Nelder&Mead Simplex search. The performance of these search algorithms' implementations was assessed using the BBOB2009 (Black Box Optimization Benchmark) benchmark suite. These algorithms were compared in the context of two optical case studies, one with conventional rotationally symmetric optics and one with freeform optics. A comparison was performed against a commercial optical design software. Finally we provided a simple restart scheme applicable in the workflow of an optical designer. To the best of our knowledge, this is the first in-depth quantitative comparison of optimization algorithms for optical design.

Light in power

We present in this paper a generic and parameter-free algorithm to efficiently build a wide variety of optical components, such as mirrors or lenses, that satisfy some light energy constraints. In all of our problems, one is given a collimated or point light source and a desired illumination after reflection or refraction and the goal is to design the geometry of a mirror or lens which transports exactly the light emitted by the source onto the target. We first propose a general framework and show that eight different optical component design problems amount to solving a light energy conservation equation that involves the computation of visibility diagrams. We then show that these diagrams all have the same structure and can be obtained by intersecting a 3D Power diagram with a planar or spherical domain. This allows us to propose an efficient and fully generic algorithm capable to solve these eight optical component design problems. The support of the prescribed target illumination can be a set of directions or a set of points located at a finite distance. Our solutions satisfy design constraints such as convexity or concavity. We show the effectiveness of our algorithm on simulated and fabricated examples.

Fast, deterministic computation of irradiance values using a single extended source in 3D.

Presented is a quasi-analytic method for irradiance evaluation through a single refractive surface from a single Lambertian source. The method is compared to Monte-Carlo raytracing for a sample system, producing in much less time an irradiance distribution equal to within the latter's statistical noise. In addition to its interest to optical analysis, the method is also useful for optical design problems by allowing for fast, noise-free evaluation of merit functions, along with their derivatives.

Reflecting anastigmatic optical systems: a retrospective

Reflecting anastigmatic optical systems hold several inherent advantages over refracting equivalents, such as compactness, absence of color, high “refractive efficiency,” wide bandwidth, and size scalability to enormous apertures. Such advantages have led to these systems becoming, increasingly since their first deliberate development in 1905, the “go-to” solution for various classes of optical design problems. This paper describes in broad terms the history of the development of this class of optical system, with an emphasis on the early history.

The Design of Transparent Optical Networks Minimizing the Impact of Critical Nodes

For a given fiber network and a given set of client demands, the transparent optical network design problem is the task of assigning routing paths and wavelengths for a set of lightpaths able to groom all client demands. We address this design problem minimizing the impact of a given set of critical nodes. The problem is tackled in two steps: first, we minimize the demand that is disrupted by the simultaneous failure of all critical nodes; second, we minimize the network design cost guaranteeing that the minimum disrupted demand is met. We present MILP models for each step, together with valid inequalities strengthening both models. For the second step, an efficient hybrid heuristic is also proposed.

Emergency optical network planning with multi-vendor interconnection and portable EDFAs

In this paper, we introduce an emergency optical network design problem for the low-cost post-disaster recovery of core optical transport networks. We take into account both the interconnection of the surviving multi-vendor resources and the utilization of portable emergency erbium-doped fiber amplifiers (EDFAs). The surviving multi-vendor optical nodes (e.g., reconfigurable optical add/drop multiplexer (ROADM) or wavelength cross-connect (WXC) nodes) and fiber links are the available resources during a disaster recovery and should be efficiently utilized first. The portable EDFAs are emergency resources added to replace the optical nodes that have either been destroyed or are down because of power outages. To solve this design problem, we propose an emergency network planning method using an integer linear programming (ILP) formulation such that the locations for the emergency interconnection of the multi-vendor networks and placement of the portable EDFAs are optimally selected. Evaluations show the benefits of the multi-vendor interconnection approach and utilization of emergency portable EDFAs.

The Minimum Cost Design of Transparent Optical Networks Combining Grooming, Routing, and Wavelength Assignment

As client demands grow, optical network operators are required to introduce lightpaths of higher line rates in order to groom more demand into their network capacity. For a given fiber network and a given set of client demands, the minimum cost network design is the task of assigning routing paths and wavelengths for a minimum cost set of lightpaths able to groom all client demands. The variant of the optical network design problem addressed in this paper considers a transparent optical network, single hop grooming, client demands of a single interface type, and lightpaths of two line rates. We discuss two slightly different mixed integer linear programming models that define the network design problem combining grooming, routing, and wavelength assignment. Then, we propose a parameters increase rule and three types of additional constraints that, when applied to the previous models, make their linear relaxation solutions closer to the integer solutions. Finally, we use the resulting models to derive a hybrid heuristic method, which combines a relax-and-fix approach with an integer linear programming-based local search approach. We present the computational results showing that the proposed heuristic method is able to find solutions with cost values very close to the optimal ones for a real nation-wide network and considering a realistic fiber link capacity of 80 wavelengths. Moreover, when compared with other approaches used in the problem variants close to the one addressed here, our heuristic is shown to compute solutions, on average, with better cost values and/or in shorter runtimes.

Benders Decomposition of the Passive Optical Network Design Problem

Similar to the connected facility location problem, the passive optical network design problem requires the search for a subset of deployed distribution points (splitters) as well as an allocation of demand points (optical network units) to minimise deployment cost. In this paper we decompose a path-based relaxation of the problem using Benders as well as column generation and analyse strengthening cuts for the resulting master. Computational results for this approach are then illustrated.

Superdefect Photonic Crystal Filter Optimization Using Grey Wolf Optimizer

This letter proposes a new method for designing photonic crystal (PhC) filters. The method is proposed while designing novel superdefect PhC filters with three and five ellipse-shaped rods. The main idea of the method is to formulate the optical filter design problem as an optimization problem and to solve it by single-objective optimization techniques. This method bypasses discovering the complex relationship between the output spectral transmission performance and the radii of rods. The results demonstrate the merits of the method in facilitating the design process of the PhC filters and finding high-performance designs.