Modelling Real Valued Functions via Optical Lenses

Year 2024, , 63 - 70, 30.08.2024

Furkan Semih Dündar

https://doi.org/10.30931/jetas.1312369

Abstract

In this study, we modeled real valued functions using freeform lenses. In our model, the bottom surface of the lens is flat whereas its top surface is determined by a function, f(x). We consider vertically coming light rays with x-coordinate x. Our aim is to find f(x) such that x is mapped to F(x), the horizontal position where the light ray leaves the bottom surface. We have found the nonlinear differential equation for a generic lens to model a given function. Namely, given F(x), the solution of the differential equation gives us the lens surface f(x). Finally, we have calculated the lens surface for four functions numerically and have provided their plots respectively.

Keywords

Freeform lens design, ray optics, mathematical modelling

References

• [1] Doskolovich, L.L., Mingazov, A.A., Bykov, D.A., Bezus, E.A., “Formulation of the inverse problem of calculating the optical surface for an illuminating beam with a plane wavefront as the Monge Kantrovich problem”, Computer Optics 43(5) (2019) : 705-713.
• [2] Fang, F.Z., Zhang, X.D., Weckenmann, A., Zhang, G.X., Evans, C., “Manufacturing and measurement of freeform optics”, CIRP Annals 62(2) (2013) : 823-846.
• [3] Wolfram Research, Inc. Mathematica, Version 13.2. Champaign, IL. (2022)
• [4] Kumar, S., Tong, Z., Jiang, X., “Advances in the design and manufacturing of novel freeform optics”, International Journal of Extreme Manufacturing 4(3) (2022) : 032004.
• [5] Valencia-Estrada, J.C., Garcia-Marquez, J., “Freeform geometrical optics i: principles”, Applied Optics 58(34) (2019) : 9455-9464.
• [6] van Roosmalen, A.H., Anthonissen, M.J.H., Ijzerman, W.L., ten Thije Boonkkamp, J.H.M., “Fresnel reflections in inverse double freeform lens design”, Journal of the Optical Society of America A 40(7) (2023) : 1310-1318.
• [7] Wu, R., Chang, S., Zheng, Z., Zhao, L., Liu, X., “Formulating the design of two freeform lens surfaces for point-like light sources.” Optics Letters 43(7) (2018) : 1619-1622. [8] Yang, L., Liu, Y., Ding, Z., Zhang, J, Tao, X., Zheng, Z.R., Wu, R., “Design of freeform lenses for illuminating hard-to-reach areas through a light-guiding system”, Optics Express 28(25) (2020) : 38155-38168.