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Journals >Chinese Optics Letters >Volume 21 >Issue 5 >Page 052201 > Article

Chinese Optics Letters
Vol. 21, Issue 5, 052201 (2023)

Parallel ray tracing through freeform lenses with NURBS surfaces

Haisong Tang^1、2, Zexin Feng^1、2、*, Dewen Cheng^1、2, and Yongtian Wang^1、2

Author Affiliations

¹Beijing Engineering Research Center of Mixed Reality and Advanced Display, School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China

²MOE Key Laboratory of Optoelectronic Imaging Technology and Systems, Beijing Institute of Technology, Beijing 100081, China

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DOI: 10.3788/COL202321.052201 Cite this Article Set citation alerts

Haisong Tang, Zexin Feng, Dewen Cheng, Yongtian Wang. Parallel ray tracing through freeform lenses with NURBS surfaces[J]. Chinese Optics Letters, 2023, 21(5): 052201 Copy Citation Text

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Abstract

We implement Monte Carlo-based parallel ray tracing to achieve quick irradiance evaluation for freeform lenses with non-uniform rational B-splines (NURBS) surfaces. We employ the inverse transform sampling method to sample rays uniformly from the Lambertian light source and adopt the analytical form of the B-spline basis function to achieve fast surface interpolation. When performing parallel calculations for the intersections between the rays and the NURBS surfaces, we propose a parameter transformation method to avoid the parameters escaping from the defined range in the iteration process. Simulation results of two complex picture-generating freeform lenses show that our method is fast and effective.

Keywords

computation methods lenses nonimaging optics propagation methods

L (x, y, r_{x}, r_{y}) \cdot \cos θ δ S δ Ω \propto E_{μ} (\sum_{i = 0}^{k} w_{i}) .

(1)

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Q (u, v) = \sum_{i = 0}^{m_{s} - 1} \sum_{j = 0}^{n_{s} - 1} N_{i, p} (u) N_{j, q} (v) R_{i, j} P (i, j),

(2)

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{\begin{cases} N_{i,0} = {\begin{cases} 1 & if u \in [u_{i}, u_{i + 1}) \\ 0 & if else \end{cases}, \\ N_{i, p} = \frac{u - u_{i}}{u_{i + p} - u_{i}} N_{i, p - 1} (u) + \frac{u_{i + p + 1} - u}{u_{i + p + 1} - u_{i + 1}} N_{i + 1, p - 1} (u), \\ define: \frac{0}{0} = 0 . \end{cases}

(3)

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Q (u, v) = \hat{s} + \hat{r} \cdot t,

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{\begin{cases} Q_{1} (u, v) - x_{0} = \sin θ \cos φ \cdot t \\ Q_{2} (u, v) - y_{0} = \sin θ \sin φ \cdot t \\ Q_{3} (u, v) - z_{0} = \cos θ \cdot t \end{cases} .

(5)

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R_{z} = [\begin{matrix} \cos φ & \sin φ & 0 \\ - \sin φ & \cos φ & 0 \\ 0 & 0 & 1 \end{matrix}], R_{y} = [\begin{matrix} \cos θ & 0 & - \sin θ \\ 0 & 1 & 0 \\ \sin θ & 0 & \cos θ \end{matrix}],

(6)

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R_{y} \cdot R_{z} \cdot [\begin{matrix} Q_{x} (u, v) - x_{0} \\ Q_{y} (u, v) - y_{0} \\ Q_{z} (u, v) - z_{0} \end{matrix}] = R_{y} \cdot R_{z} \cdot [\begin{matrix} \sin θ \cdot \cos φ \cdot t \\ \sin θ \cdot \sin φ \cdot t \\ \cos θ \cdot t \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ t \end{matrix}] .

(7)

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[\begin{matrix} u_{k + 1} \\ v_{k + 1} \end{matrix}] = [\begin{matrix} u_{k} \\ v_{k} \end{matrix}] - J^{- 1} [\begin{matrix} f_{1} (u_{k}, v_{k}) \\ f_{2} (u_{k}, v_{k}) \end{matrix}],

(8)

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J = [\begin{matrix} \frac{\partial f_{1}}{\partial u} & \frac{\partial f_{1}}{\partial v} \\ \frac{\partial f_{2}}{\partial u} & \frac{\partial f_{2}}{\partial v} \end{matrix}] .

(9)

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u = \frac{1}{e^{- 4 α} + 1}, v = \frac{1}{e^{- 4 β} + 1} .

(10)

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{\begin{cases} α, β \in (- \infty, + \infty), u, v \in (0, 1) \\ \frac{d u}{d α} > 0, \frac{d v}{d β} > 0, {\frac{d u}{d α} |}_{α = 0} = 1, {\frac{d v}{d β} |}_{β = 0} = 1 \\ {u |}_{α = 0} = 0.5, {v |}_{β = 0} = 0.5 \end{cases} .

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{\begin{matrix} \frac{\partial f}{\partial α} & = & \frac{\partial f}{\partial u} \cdot \frac{d u}{d α} \\ \frac{\partial f}{\partial β} & = & \frac{\partial f}{\partial v} \cdot \frac{d v}{d β} \end{matrix} .

(12)

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[\begin{matrix} α_{k + 1} \\ β_{k + 1} \end{matrix}] = [\begin{matrix} α_{k} \\ β_{k} \end{matrix}] - J^{- 1} [\begin{matrix} f_{1} (α_{k}, β_{k}) \\ f_{2} (α_{k}, β_{k}) \end{matrix}] .

(13)

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\hat{t} = \frac{1}{n_{2}} {n_{1} \hat{i} + [\sqrt{n_{2}^{2} - n_{1}^{2} + n_{1}^{2} {(\hat{n} \cdot \hat{i})}^{2}} - n_{1} \hat{n} \cdot \hat{i}] \hat{n}} .

(14)

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r_{s} = \frac{n_{1} \cos θ_{i} - n_{2} \cos θ_{t}}{n_{1} \cos θ_{i} + n_{2} \cos θ_{t}}, r_{p} = \frac{n_{1} \cos θ_{t} - n_{2} \cos θ_{i}}{n_{1} \cos θ_{t} + n_{2} \cos θ_{i}},

(15)

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W_{unit} = \frac{W_{0}}{\sum_{i = 1}^{n} w_{0, i}} .

(16)

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S_{cell} = \frac{(x_{\max} - x_{\min}) \cdot (y_{\max} - y_{\min})}{m_{c} \cdot n_{c}} .

(17)

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E_{j} = \frac{W_{unit} \times \sum_{i = 1}^{n_{j}} w_{i^{'}}}{S_{cell}} .

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References (42)

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Haisong Tang, Zexin Feng, Dewen Cheng, Yongtian Wang. Parallel ray tracing through freeform lenses with NURBS surfaces[J]. Chinese Optics Letters, 2023, 21(5): 052201

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