J.H.M. Ten Thije Boonkkamp

J.H.M. Ten Thije Boonkkamp
Eindhoven University of Technology | TUE · Department of Mathematics and Computer Science

About

207
Publications
33,274
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
2,079
Citations

Publications

Publications (207)
Article
We construct novel flux approximation schemes for the semidiscretized incompressible Navier–Stokes equations by finite‐volume method on a staggered mesh. The calculation of the cell‐face fluxes has been done by solving appropriate local non‐linear boundary value problems (BVP). Consequently, the cell‐face fluxes are represented as the sum of a homo...
Article
Full-text available
Imaging systems are inherently prone to aberrations. We present an optimization method to design two-dimensional (2D) freeform reflectors that minimize aberrations for various parallel ray beams incident on the optical system. We iteratively design reflectors using inverse methods from non-imaging optics and optimize them to obtain a system that pr...
Preprint
Full-text available
In this paper, we discuss a mathematical model for inverse freeform design of an optical system with two reflectors in which light transfers from a point source to a point target. In this model, the angular light intensity emitted from the point source and illuminance arriving at the point target are specified by distributions. To determine the opt...
Preprint
Full-text available
We present a systematic derivation of three mathematical models of increasing complexity for optical design, based on Hamilton's characteristic functions and conservation of luminous flux, and briefly explain the connection with the mathematical theory of optimal transport. We outline several iterative least-squares solvers for our models and demon...
Article
Full-text available
In this paper, we propose a numerical scheme for fluid models of magnetised plasmas. One important feature of the numerical scheme is that it should be able to handle the anisotropy induced by the magnetic field. In order to do so, we propose the use of the hybrid mimetic mixed (HMM) scheme for diffusion. This is combined with a hybridised variant...
Article
Full-text available
We present a novel approach to computing reflectors with a scattering surface in illumination optics. A scattering model governed by a Fredholm integral equation is derived. Solving this integral relation yields a virtual specular target distribution, which we insert into a Monge-Ampère least-squares numerical solver to get a scattering reflector t...
Article
Full-text available
Concatenated backward ray mapping is an alternative for ray tracing in 2D. It is based on the phase space descrip tion of an optical system. Phase space is the set of position and direction coordinates of rays intersecting an optical line. The original algorithm is limited to optical systems consisting of only straight line segments; we extend it t...
Article
Full-text available
We present a novel approach to minimize aberrations in imaging systems. The energy distributions at the source and target of an optical system play a crucial role in designing freeform surfaces through illumination optics methodologies. We quantify the on-axis and off-axis aberrations using a merit function that depends on the energy distributions....
Article
Full-text available
We introduce an approach to calculating three-dimensional freeform reflectors with a scattering surface. Our method is based on optimal transport and utilizes a Fredholm integral equation to express scattering. By solving this integral equation through a process analogous to deconvolution, we can recover a typical specular design problem. Consequen...
Article
Full-text available
In this paper we present a method for designing a double freeform lens that includes the effect of Fresnel reflections on the output intensity. We elaborate this method for the case of a point source and a far-field target. A new expression for the transmittance through a double freeform lens is derived, and we adapt a least-squares algorithm to ac...
Article
Full-text available
We apply the Lie algebraic method to reflecting optical systems with plane-symmetric freeform mirrors. Using analytical ray-tracing equations, we construct an optical map. The expansion of this map gives us the aberration coefficients in terms of initial ray coordinates. The Lie algebraic method is applied to treat aberrations up to arbitrary order...
Preprint
Full-text available
We introduce a novel approach to calculating three-dimensional freeform reflectors with a scattering surface. Our method is based on optimal transport and utilizes a Fredholm integral equation to express scattering. By solving this integral equation through a process similar to deconvolution, which we call `unfolding,' we can recover a typical spec...
Article
Full-text available
We present a novel approach of modelling surface light scattering in the context of two-dimensional reflector design, relying on energy conservation and optimal transport theory. For isotropic scattering in cylindrically or rotationally symmetric systems with in-plane scattering, the scattered light distribution can be expressed as a convolution be...
Preprint
Full-text available
A least-squares method for solving the hyperbolic Monge-Amp\`ere equation with transport boundary condition is introduced. The method relies on an iterative procedure for the gradient of the solution, the so-called mapping. By formulating error functionals for the interior domain, the boundary, both separately and as linear combination, three minim...
Article
Full-text available
We combine two-dimensional freeform reflector design with a scattering surface modeled using microfacets, i.e., small, specular, surfaces representing surface roughness. The model resulted in a convolution integral for the scattered light intensity distribution, which yields an inverse specular problem after deconvolution. Thus, the shape of a refl...
Preprint
Full-text available
We apply the Lie algebraic method to reflecting optical systems with plane-symmetric freeform mirrors. Using analytical ray-tracing equations we construct an optical map. The expansion of this map gives us the aberration coefficients in terms of initial ray coordinates. The Lie algebraic method is applied to treat aberrations up to the desired orde...
Preprint
Full-text available
We apply the Lie algebraic method to reflecting optical systems with plane-symmetric freeform mirrors. Using analytical ray-tracing equations we construct an optical map. The expansion of this map gives us the aberration coefficients in terms of initial ray coordinates. The Lie algebraic method is applied to treat aberrations up to the desired orde...
Preprint
Full-text available
We apply the Lie algebraic method to reflecting optical systems with plane-symmetric freeform mirrors. Using analytical ray-tracing equations we construct an optical map. The expansion of this map gives us the aberration coefficients in terms of initial ray coordinates. The Lie algebraic method is applied to treat aberrations up to arbitrary order....
Chapter
Liouville’s equation describes light propagation through an optical system. It governs the evolution of an energy distribution on phase space. This distribution is discontinuous across optical interfaces. The discontinuous Galerkin spectral element method is employed to solve Liouville’s equation. At optical interfaces the laws of optics describe n...
Preprint
Full-text available
We combine two-dimensional freeform reflector design with a scattering surface modelled using microfacets, i.e., small specular surfaces representing surface roughness. The model results in a convolution integral for the scattered light intensity distribution, which yields an inverse specular problem after deconvolution. Thus, the shape of a reflec...
Article
Full-text available
The Lie algebraic method offers a systematic way to find aberration coefficients of any order for plane-symmetric reflective optical systems. The coefficients derived from the Lie method are in closed form and solely depend on the geometry of the optical system. We investigate and verify the results for a single reflector. The concatenation of mult...
Article
Full-text available
We present a method for designing freeform optical surfaces for illumination optics. By the laws of reflection, refraction and conservation of energy, a fully nonlinear PDE, the Monge-Ampere equation, is derived for the optical surface. By the edge ray principle a transport boundary condition is obtained. We solve the hyperbolic variant of the PDE...
Article
Full-text available
We present an inverse method for optical design that compensates local Fresnel reflections. We elaborate this method for a point source and far-field target. We modify an existing design algorithm based on the least-squares method. This is done in such a way that the shape of the transmitted intensity is as desired.
Article
Full-text available
We present an alternative method to ray tracing that is based on a phase space description of light propagation. Liouville’s equation of geometrical optics describes the evolution of the basic luminance on phase space. At an optical interface, the laws of optics describe non-local boundary conditions for the basic luminance. A discontinuous Galerki...
Article
Full-text available
We give a brief introduction to Hamiltonian optics and Lie algebraic methods. We use these methods to describe the operators governing light propagation, refraction, and reflection in phase space. The method offers a systematic way to find aberration coefficients of any order for arbitrary rotationally symmetric optical systems. The coefficients fr...
Article
Full-text available
We present three alternative derivations of the method of characteristics (MOC) for a second order nonlinear hyperbolic partial differential equation (PDE) in two independent variables. The MOC gives rise to two mutually coupled systems of ordinary differential equations (ODEs). As a special case we consider the Monge–Ampère (MA) equation, for whic...
Preprint
Full-text available
We give a brief introduction to Hamiltonian optics and Lie algebraic methods. We use these methods to describe the operators governing light propagation, refraction and reflection in phase space. The method offers a systematic way to find aberration coefficients of any order for arbitrary rotationally symmetric optical systems. The coefficients fro...
Article
Full-text available
In this paper we propose a method to design a freeform lens including the effect of Fresnel reflections on the transmitted intensity. This method is elaborated for a lens with one freeform surface shaping a far-field target from a point source or collimated input beam. It combines the optical mapping with the energy balance incorporating the loss d...
Preprint
Full-text available
In this work, we consider an advection-diffusion equation, coupled to a Poisson equation for the velocity field. This type of coupling is typically encountered in models arising from plasma physics or porous media flow. The aim of this work is to build upon the complete flux scheme (an improvement over the Scharfetter-Gummel scheme by considering t...
Preprint
Full-text available
In this paper we propose a method to design a freeform lens including the effect of Fresnel reflections on the transmitted intensity. This method is elaborated for a lens with one freeform surface shaping a far-field target from a point source or collimated input beam. It combines the optical mapping with the energy balance incorporating the loss d...
Article
Full-text available
A new method to compute the target photometric variables of non-imaging optical systems is presented. The method is based on the phase space representation of each surface that forms the optical system. All surfaces can be modeled as detectors of the incident light and emitters of the reflected light. Moreover, we assume that the source can only em...
Article
Full-text available
Liouville’s equation on phase space in geometrical optics describes the evolution of an energy distribution through an optical system, which is discontinuous across optical interfaces. The discontinuous Galerkin spectral element method is conservative and can achieve higher order of convergence locally, making it a suitable method for this equation...
Article
Full-text available
We present a unified mathematical framework for sixteen fundamental optical systems. The systems have a parallel or point source and a parallel, point, near-field or far-field target. These choices give eight configurations if we use reflectors only and take the minimum number of freeform surfaces required. Similarly, we get eight lens systems if w...
Article
Full-text available
In this paper we propose a method to compute a freeform reflector system for collimating and shaping a beam from a point source. We construct these reflectors such that the radiant intensity of the source is converted into a desired target. An important generalization in our approach compared to previous research is that the output beam can be in a...
Preprint
Full-text available
In this paper, we propose a numerical scheme for fluid models of magnetised plasmas. One important feature of the numerical scheme is that it should be able to handle the anisotropy induced by the magnetic field. In order to do so, we propose the use of the hybrid mimetic mixed (HMM) scheme for diffusion. This is combined with a hybridised variant...
Preprint
Full-text available
We present a novel approach of modelling surface light scattering in the context of freeform optical design. The model relies on energy conservation and optimal transport theory. For isotropic scattering in cylindrically or rotationally symmetric systems with in-plane scattering, the scattered light distribution can be expressed as a convolution be...
Preprint
Full-text available
We present three alternative derivations of the method of characteristics (MOC) for a second order nonlinear hyperbolic partial differential equation. The MOC gives rise to two mutually coupled systems of ordinary differential equations. As a special case we consider the Monge-Amp\`ere equation, for which we solve the system of ODE's using explicit...
Preprint
Full-text available
In this paper we propose a method to compute a freeform reflector system for collimating and shaping a beam from a point source. We construct these reflectors such that the radiant intensity of the source is converted into a desired target. An important generalization in our approach compared to previous research is that the output beam can be in a...
Article
Full-text available
In this paper, we consider separating the discretisation of the diffusive and advective fluxes in the complete flux scheme. This allows the combination of several discretisation methods for the homogeneous flux with the complete flux (CF) method. In particular, we explore the combination of the hybrid mimetic mixed (HMM) method and the CF method, i...
Article
Full-text available
Many LED lighting applications involve the design of multiple optical surfaces. A prime example is a single lens with two refractive surfaces. In this paper, we consider an LED light source approximated as a point and a far-field target intensity. Using Hamilton’s characteristic functions, the design problem is converted into two generalized Monge–...
Chapter
Designing freeform optical surfaces that control the redistribution of light from a particular source distribution to a target irradiance poses challenging problems in the field of illumination optics. There exists a wide variety of strategies in academia and industry, and there is an interesting link with optimal transport theory. Many freeform op...
Article
Full-text available
We present a method to design a freeform two-reflector system to collimate and shape a beam from a point source. An important generalization compared to previous research is that the output beam can be in an arbitrary direction. The design problem is based on a generalized Monge-Ampère equation. This equation is solved using a least-squares algorit...
Article
Full-text available
We present a novel approach of modelling surface light scattering in the context of freeform optical design. Using energy conservation, we derive an integral relation between the scattered and specular distributions. This integral relation reduces to a convolution integral in the case of isotropic scattering in the plane of incidence for cylindrica...
Conference Paper
We outline the mathematical framework for freeform optical design, based on Hamilton’s characteristic functions and energy conservation.
Conference Paper
In this invited talk, I will present the mathematics of freeform systems with point sources using the theory of generated Jacobian equations, derived using Hamilton’s characteristic functions. We use an efficient least-squares numerical procedure.
Article
Full-text available
In this contribution an alternative method to standard forward ray-tracing is briefly outlined. The method is based on a phase-space description of light propagating through an optical system. The propagation of light rays are governed by Hamilton’s equations. Conservation of energy and étendue for a beam of light, allow us to derive a Liouville’s...
Chapter
We present the derivation of the generalized Monge–Ampère equation for two optical systems, viz. a freeform lens with parallel incident and refracted light rays, which transforms a source emittance into a desired target illuminance, and a freeform reflector converting the intensity of a point source into a far-field distribution. The derivations ar...
Chapter
The physical modeling of transport in multi-component mixtures results in systems of coupled equations for the mass fractions. This contribution discusses the mathematical structure of such transport systems and presents a novel approximation scheme for the associated mass fluxes. The scheme respects the coupled nature of the equations and allows f...
Article
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 sha...
Article
Full-text available
In this paper, we present a comparison of two novel exponential schemes for convection-diffusion problems. An exponential scheme uses in one way or another the analytical solution of the flux of a one-dimensional (1D) transport equation thereby improving the results of the simulation. In a multidimensional problem, the 1D solution is combined with...
Article
Full-text available
We present an initial attempt to model scattering in freeform optical design. Scattering is modelled as a convolution of the unperturbed specular distribution and a spreading function. Deconvolution is used to recover the equivalent specular distribution, for which design procedures are well established.
Article
Full-text available
We present an efficient numerical algorithm that can be used to solve the generalized Monge-Ampère equations for a single freeform reflector and lens surface. These equations are instances of so-called ‘generated Jacobian equations’ which are characterized by associated generating functions. The algorithm has a wide applicability to any optical sys...
Article
Full-text available
The field of freeform illumination design has surged since the introduction of new fabrication techniques that allow for the production of non-axially symmetric surfaces. Freeform surfaces aim to efficiently control the redistribution of light from a particular source distribution to a target irradiance, but designing such surfaces is a challenging...
Article
Full-text available
The purpose of this paper is to present a method for the design of two-reflector optical systems that transfer a given energy density of the source to a desired energy density at the target. It is known that the two-reflector design problem gives rise to a Monge–Ampère (MA) equation with transport boundary condition. We solve this boundary value pr...
Article
Full-text available
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...
Article
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 sur...
Article
Full-text available
Active flux schemes are finite volume schemes that keep track of both point values and averages. The point values are updated using a semi-Lagrangian step, making active flux schemes highly suitable for geometric optics problems on phase space, i.e., to solve Liouville's equation. We use a semi-discrete version of the active flux scheme. Curved opt...
Article
In this study, we propose a new scheme named as complete flux scheme (CF-scheme) based on the finite volume method for solving singularly perturbed differential–difference equations (SPDDEs) of elliptic type. An alternate integral representation for the flux is obtained which plays an important role in the derivation of CF-scheme. We have establish...
Preprint
The purpose of this paper is to present a method for the design of two-reflector optical systems that transfer a given energy density of the source to a desired energy density at the target. It is known that the two-reflector design problem gives rise to a Monge-Ampère equation with transport boundary condition. We solve this boundary value problem...
Preprint
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 wave-fronts. 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 su...
Preprint
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...
Chapter
We present a novel flux approximation scheme for the viscous Burgers equation. The numerical flux is computed from a local two-point boundary value problem for the stationary equation and requires the iterative solution of a nonlinear equation depending on the local boundary values and the viscosity. In the inviscid limit the scheme reduces to the...
Chapter
We aim to solve inverse problems in illumination optics by means of optimal control theory. This is done by first formulating geometric optics in terms of Liouville’s equation, which governs the evolution of light distributions on phase space. Choosing a metric that measures how close one distribution is to another, the formal Lagrange method can b...
Conference Paper
Full-text available
Freeform optical surfaces can transfer a given light distribution of the source into a desired distribution at the target. Freeform optical design problems can be formulated as a Monge-Ampère type differential equation with transport boundary condition, using properties of geometrical optics, conservation of energy, and the theory of optimal mass t...
Conference Paper
A method is presented for the design of a freeform lens for a point source and far-field target. We use a least-squares algorithm to solve the generalized Monge-Ampère equation for a few test problems.