Lens Design Problem Research Articles

Revisiting a classic lens design problem

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.

Design Ultra-Compact Aspherical Lenses for Extended Sources Using Particle Swarm Optical Optimization Algorithm

This paper presents a global optimization algorithm specifically tailored for ultra-compact aspherical lens design problems for extended LED sources. The main purpose is to obtain prescribed illumination patterns, particularly the uniform illuminance distribution. This method begins by calculating the initial aspherical lens with two surfaces based on point source approximation. Then a system of polynomials is employed to fit the meridian curves of the two surfaces. In the optimization process, we use the particle swarm optimization (PSO) algorithm to automatically find the optimal polynomial coefficients when the ray tracing simulation is being performed. A series of ultra-compact aspherical lenses with dimension ratio of h / d ranging from 0.95 to 1.25 are presented, where h denotes the center height of the lens and d represents the diameter of the extended source. The results show the high efficiency and versatility of the proposed method in prescribed illumination design for extended LED sources in three-dimensional rotational symmetry geometry. Additionally, an aspherical lens is fabricated and tested, and its practical performance approaches the design.

Transferring freeform lens design into phase retrieval through intermediate irradiance transport.

Freeform lens design for the prescribed irradiance without thin element and paraxial approximations is very complicated. We propose to transport a point source irradiance before the required freeform lens into an intermediate irradiance estimate next to the lens. In this way, the complicated freeform lens design problem could be transferred into a simpler problem of retrieving the phase from the intermediate and target irradiances. The phase retrieval solution can help construct an approximate lens, which is subsequently used to update the intermediate irradiance. Several iterations could bring a good performance, as demonstrated by two challenging designs.

A Monge\u2013Amp\xe8re Problem with Non-quadratic Cost Function to Compute Freeform Lens Surfaces

In this article, we present a least-squares method to compute freeform surfaces of a lens with parallel incoming and outgoing light rays, which is a transport problem corresponding to a non-quadratic cost function. The lens can transfer a given emittance of the source into a desired illuminance at the target. The freeform lens design problem can be formulated as a Monge–Ampère type differential equation with transport boundary condition, expressing conservation of energy combined with the law of refraction. Our least-squares algorithm is capable to handle a non-quadratic cost function, and provides two solutions corresponding to either convex or concave lens surfaces.

Computation of double freeform optical surfaces using a Monge–Ampère solver: Application to beam shaping

In this article, we present a formulation for the design of double freeform lens surfaces to control the intensity distribution of a laser beam with plane wavefronts. Double freefrom surfaces are utilized to shape collimated beams. Two different layouts of the freeform lens optical system are introduced, i.e., a single lens with double freeform surfaces, and two separate lenses with two flat and two freeform surfaces. The freeform lens design problem can be formulated as a Monge–Ampère type differential equation with transport boundary condition, expressing conservation of energy combined with the law of refraction and the constraint imposed on the optical path length between source and target planes. Numerical solutions are computed using a generalized least-squares algorithm which is presented by Yadav et al. (2018). The algorithm is capable to compute two solutions of the Monge–Ampère boundary value problem, corresponding to either c-convex or c-concave freeform surfaces for both layouts. The freeform surfaces are validated for several numerical examples using a ray-tracer based on Quasi-Monte Carlo simulation.

Open-source optimization algorithms for optical design

Driven by the growth of internet, commercial cloud computing now provides on-demand, massively parallel computational power. This cloud processing can be utilized for solving science and engineering problems. Relying on computationally intensive ray-tracing, optical design can greatly benefit from a cloud based optimization approach if suitable parallelizable optimization algorithms can be implemented. Commercial optical design packages have algorithms tuned for optical design. There are also various local and global optimization algorithms for scientific and engineering problems. In this paper, several general purpose, open-source local and global optimization algorithms are evaluated for a lens design problem. The results are compared to results from a commercial optical design package, and some algorithms are shown to give similar performance. This research provides valuable insight into different optimization algorithms and therefore is a first step for taking optical design from desktop computers to cloud computing in order to achieve massively parallel processing in system optimization.

AEGA: A New Real-Coded Genetic AlgorithmTaking Account of Extrapolation

This study proposes a new real-coded genetic algorithm (RCGA) taking account of extrapolation, which we call adaptive extrapolation RCGA (AEGA). Real-world problems are often formulated as black-box function optimization problems and sometimes have ridge structures and implicit active constraints. mAREX/JGG is one of the most powerful RCGAs that performs well against these problems. However, mAREX/JGG has a problem of search inefficiency. To overcome this problem, we propose AEGA that generates offspring outside the current population in a more stable manner than mAREX/JGG. Moreover, AEGA adapts the width of the offspring distribution automatically to improve its search efficiency. We evaluate the performance of AEGA using benchmark problems and show that AEGA finds the optimum with fewer evaluations than mAREX/JGG with a maximum reduction ratio of 45%. Furthermore, we apply AEGA to a lens design problem that is known as a difficult real-world problem and show that AEGA reaches the known best solution with approximately 25% fewer evaluations than mAREX/JGG.

Multi-objective approach for the automatic design of optical systems

An innovative method for the automatic design of optical systems is presented and verified. The proposed method is based on a multi-objective evolutionary memetic optimization algorithm. The multi-objective approach simultaneously, but separately, addresses the image quality, tolerance, and complexity of the system. The memetic technique breaks down the search for optical designs in to three different parts or phases: optical glass selection, exploration, and exploitation. The optical glass selection phase defines the most appropriate set of glasses for the system under design. The glass selection phase limits the available glasses from hundreds to just a few, drastically reducing the design space and significantly increasing the efficiency of the automatic design method. The exploration phase is based on an evolutionary algorithm (EA), more specifically, on a problem-tailored generalized extremal optimization (GEO) algorithm, named Optical GEO (O-GEO). The new EA incorporates many features customized for lens design, such as optical system codification and diversity operators. The trade-off systems found in the exploration phase are refined by a local search, based on the damped least square method in the exploitation phase. As a result, the method returns a set of trade-off solutions, generating a Pareto front. Our method delivers alternative and useful insights for the compromise solutions in a lens design problem. The efficiency of the proposed method is verified through real-world examples, showing excellent results for the tested problems.

2205 レベルセット法に基づくレンズ設計問題のトポロジー最適化

2205 レベルセット法に基づくレンズ設計問題のトポロジー最適化

Intensity control with a free-form lens

A family of free-form lenses for intensity control is designed. The lens can shape an incident collimated beam with a given intensity distribution I1 into a new collimated beam with intensity distribution I2. No symmetry is assumed for the two intensity profiles. The key idea is that the lens design problem can be formulated and solved in terms of an optimization process involving a specific action functional. It is further shown that the free-form lens can be manufactured by a surfacing process using a convex tool.

Analysis of a Variational Approach to Progressive Lens Design

We consider a variational approach to the progressive lens design problem. The corresponding Euler--Lagrange equation is a fourth-order nonlinear elliptic partial differential equation. We analyze two linearizations of the equation and show the existence and uniqueness as well as the regularity of the solutions for various boundary conditions. We end with an example of a progressive lens designed by solving the elliptic partial differential equation.

Optimal lens design by real-coded genetic algorithms using UNDX

This paper presents new lens optimization methods based on real-coded genetic algorithms (GAs). We take advantage of GA's capability of global optimization and multi-objective optimization against two serious problems in conventional lens optimization techniques: (1) choosing a starting point by trial and error, and (2) combining multiple criteria to a single criterion. In this paper, two criteria for lenses, the resolution and the distortion, are considered. First, we propose a real-coded GA that optimizes a single criterion, a weighted sum of the resolution and the distortion. To overcome a problem of the difficulty in generating feasible lenses especially in large-scale problems, we introduce a feasibility enforcement operator to modify an infeasible solution into a feasible one. By applying the proposed method to some small-scale problems, we show that the proposed method can find empirically optimal and suboptimal lenses. We also apply the proposed method to some relatively large-scale problems and show that the proposed method can effectively work under large-scale problems. Next, regarding the lens design problem as a multi-objective optimization problem, we propose a real-coded multi-objective GA that explicitly optimizes the two criteria, the resolution and the distortion. We show the effectiveness of the proposed method in multi-objective lens optimization by applying it to a three-element lens design problem.

Formulation of an Automatic Lens Design Problem by Modification of Weighted Tchebycheff Norm Method

Automatic lens design is viewed as a multiobjective optimization problem and is resolved by a minimization of the weighted Tchebycheff norm of objective functions. It is analytically solved using the multiplier method and a solution vector is transformed into the same form as the normal equation of the least squares method. Some numerical examples are given to show the effectiveness of the proposed method. The resultant solutions can not be obtained using the damped least squares method.

Determination of Initial Variable Derivative Increments Using Genetic Algorithm in Damped Least-Squares Automatic Lens Design Problem

An important factor in performing effective optimization with the damped least-squares method is to establish appropriate initial values for the variable derivative increments prior to starting the optimization process. It is shown first that the determination of these increments can be treated as a combinatorial problem. Then, a novel method of determining optimum variable derivative increments is developed using a genetic algorithm and the characteristics of the eigenvalues of the Jacobian matrix. Some numerical experiments to show the effectiveness of this method are also presented. The proposed method reduces the number of optimization reiterations required to reach a stationary point.

Can the bounds to system performance in geometrical optics be attained?

The laws of geometrical optics establish that certain imaging properties cannot be realized. For example, it is impossible to image sharply any three-dimensional region with nonunit magnification. Hamilton’s characteristic functions not only offer a simple means to determine whether certain properties are unattainable but have also been used to derive bounds in answer to the more practical question: “How closely can any particular unattainable ideal be approached?” Basic matters that relate to these bounds and whether they can be attained by realizable systems are considered here. Limits to the task of imaging more than a single plane object are used for illustration. A fundamental constraint has been overlooked in earlier research on this topic and a new bound is derived for the performance of systems in a classic lens design problem, but new questions emerge and remain unanswered.

Method for estimating numerical adequacy of derivative increments of variables in the damped least-squares automatic lens design problem

It is shown that an analytical approach for setting an adequate initial value of a damping factor in the damped least-squares optimization process can also be used to estimate whether the given increments of variables are adequate together with some numerical experiments. It is clarified that the value of a damping factor should almost equal the median of a series of eigenvalues of a squared Jacobian matrix, which is evaluated by use of given increments.

Formulation of a constrained automatic lens design problem by means of the augmented Lagrangian function

It is formulated a constrained nonlinear automatic lens design problem by means of the augmented Lagrangian function. By introducing transformation matrices, the solution is transformed into the normal equation of a linear least squares problem.

Formulation of a nonlinear automatic lens design problem with boundary conditions by means of the Kuhn-Tacker's optimality condition

Presents a constrained nonlinear automatic lens design problem formulated with the Kuhn-Tucker's optimality conditions. At first, the first order solutions are given, then the second order solutions are presented together with their formal linearization process.

GLOBAL OPTIMIZATION IN LENS DESIGN

The demand for lens systems of unconventional forms, coupled with the increased computing power available to most lens designers, has motivated the development of global optimization algorithms for lens design. The successful implementation of such an algorithm requires both a well designed global optimization scheme and an understanding of the special requirements posed by lens design problems. The simulated annealing algorithm for global optimization has attracted considerable attention from lens designers; existing variants of simulated annealing, however, lack the desired invariance under linear transformations of the coordinates or of the objective function, and typically require the specification of several hand-picked, problem-specific parameters. Success in applying these methods to lens design problems has therefore been limited. A new variant of simulated annealing-Adaptive Simulated Annealing (ASA)-is presented here in the general context of global optimization over continuous variables subject to constraints. In ASA, fixed temperature schedules and step generation routines have been replaced with heuristic-based algorithms that require minimal user-specified information and that are invariant under linear transformations of either the coordinates or the objective function. A parallel-processing implementation of ASA that provides increased efficiency is presented and applied to two standard problems for illustration and comparison. The application of ASA to lens design is examined. It is found that the lens design problem as typically posed-the canonical lens design problem-is not suited to efficient optimization with ASA. Modifications for optimization with ASA that raise efficiency-in particular, nonlinear inequality constraints and nonlinear coordinate transformations-are introduced. The effectiveness of ASA in lens design is then demonstrated through application to several standard lens design problems.

A formal linearization procedure of constrained nonlinear automatic lens design problems. II. Use of slack variables and Lagrange multipliers

For pt. I, see J. Opt. (Paris), vol.21, no.6, p.241 (1990). Presents a constrained nonlinear automatic lens design problem formulated by means of Lagrange multipliers and slack variables. The first order and the second order solutions are presented. Formal linearization of the second order solutions is also derived.