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Second-order supporting quadric method for designing freeform refracting surfaces generating prescribed irradiance distributions

Albert A. Mingazov, Dmitry A. Bykov, Evgeni A. Bezus, Leonid L. Doskolovich

Abstract

We consider the inverse problem of calculating a refracting surface that generates a prescribed irradiance distribution in the far field for a collimated incident beam. This problem can be formulated as a mass transportation problem (MTP) with a quadratic cost function. To solve this problem, we propose a version of the supporting quadric method (SQM), in which the calculation of the quadric parameters is reduced to the problem of minimizing a convex function. We obtain simple analytical expressions for the second derivatives of this function, making it possible to calculate the quadric parameters using second-order optimization methods. This allows us to refer to the proposed method as the second-order SQM. We demonstrate high efficiency of this approach by designing several optical surfaces that generate complex irradiance distributions. We also consider the application of the second-order SQM to nonimaging optics problems described by MTPs with a non-quadratic cost function.

Second-order supporting quadric method for designing freeform refracting surfaces generating prescribed irradiance distributions

Abstract

We consider the inverse problem of calculating a refracting surface that generates a prescribed irradiance distribution in the far field for a collimated incident beam. This problem can be formulated as a mass transportation problem (MTP) with a quadratic cost function. To solve this problem, we propose a version of the supporting quadric method (SQM), in which the calculation of the quadric parameters is reduced to the problem of minimizing a convex function. We obtain simple analytical expressions for the second derivatives of this function, making it possible to calculate the quadric parameters using second-order optimization methods. This allows us to refer to the proposed method as the second-order SQM. We demonstrate high efficiency of this approach by designing several optical surfaces that generate complex irradiance distributions. We also consider the application of the second-order SQM to nonimaging optics problems described by MTPs with a non-quadratic cost function.
Paper Structure (16 sections, 2 theorems, 47 equations, 7 figures, 1 table)

This paper contains 16 sections, 2 theorems, 47 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. The problem of calculating a refracting surface
  3. Direct and inverse problems
  4. Representation of the refracting surface and the mass transportation problem
  5. Supporting quadric method
  6. Calculation of the Hessian of the function $h$
  7. Numerical examples
  8. Illuminating a square region
  9. Illuminating a non-convex region and generating a grayscale image
  10. Application to nonimaging optics problems with a non-quadratic cost function
  11. Quadratic approximation of the cost function
  12. Numerical example
  13. Conclusion
  14. Supplementary materials
  15. Approximation of an MTP with an arbitrary cost function by an MTP with a quadratic cost function
  16. ...and 1 more sections

Key Result

Theorem 4.1

The second derivatives of the function $h$ have the form where $\Delta_{jk}$ is the common boundary between the cells $C^{\bm{w}}(\bm{x}_j)$ and $C^{\bm{w}}(\bm{x}_k)$, and $\rho_{jk} = \rho(\bm{x}_j, \bm{x}_k)$ is the distance between the points $\bm{x}_j$ and $\bm{x}_k$. If the cells have no common boundary, then the integral over $\Delta_{jk}$ equals z

Figures (7)

  • Figure 1: Geometry of the problem.
  • Figure 2: Polygon $W$ and increment of its area $\Delta S$ resulting from a shift of one of its sides.
  • Figure 3: Designed element (a) and generated normalized irradiance distribution (b). Cross-sections along the dashed lines are also shown in (b).
  • Figure 4: Designed element (a) and generated normalized arrow-shaped irradiance distribution in the plane $z = f = 1000\,\mathrm{mm}$. Cross-sections along the dashed lines are also shown in (b).
  • Figure 5: Designed element (a) and generated normalized grayscale irradiance distribution in the plane $z = f = 1000\,\mathrm{mm}$ (b).
  • ...and 2 more figures

Theorems & Definitions (4)

  • Theorem 4.1
  • proof
  • Theorem A.1
  • proof