Iterative freeform lens design for prescribed irradiance on curved target

Zexin Feng; Dewen Cheng; Yongtian Wang

doi:10.29026/oea.2020.200010

Journals >Opto-Electronic Advances >Volume 3 >Issue 7 >Page 200010-1 > Article

Opto-Electronic Advances
Vol. 3, Issue 7, 200010-1 (2020)

Iterative freeform lens design for prescribed irradiance on curved target

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 Tech-nology, Beijing, 100081, China

²Beijing Key Laboratory of Advanced Optical Remote Sensing Technology, School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China

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DOI: 10.29026/oea.2020.200010 Cite this Article

Zexin Feng, Dewen Cheng, Yongtian Wang. Iterative freeform lens design for prescribed irradiance on curved target[J]. Opto-Electronic Advances, 2020, 3(7): 200010-1 Copy Citation Text

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Abstract

Current freeform illumination optical designs are mostly focused on producing prescribed irradiance distributions on planar targets. Here, we aim to design freeform optics that could generate a desired illumination on a curved target from a point source, which is still a challenge. We reduce the difficulties that arise from the curved target by involving its varying z-coordinates in the iterative wavefront tailoring (IWT) procedure. The new IWT-based method is developed under the stereographic coordinate system with a special mesh transformation of the source domain, which is suitable for light sources with light emissions in semi space such as LED sources. The first example demonstrates that a rectangular flat-top illumination can be generated on an undulating surface by a spherical-freeform lens for a Lambertian source. The second example shows that our method is also applicable for producing a non-uniform irradiance distribution in a circular region of the undulating surface.

Keywords

curved target freeform optics iterative wavefront tailoring prescribed irradiance distribution

$\left\{ \begin{array}{l} u' = u\sqrt {1 - 0.5{v^2}} \\ v' = v\sqrt {1 - 0.5{u^2}} \\ \end{array} \right. .$

(1)

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$\left\{ \begin{array}{l} X = 2u'/(1 + {{u'}^2} + {{v'}^2}) \\ Y = 2v'/(1 + {{u'}^2} + {{v'}^2}) \\ Z = (1 - {{u'}^2} - {{v'}^2})/(1 + {{u'}^2} + {{v'}^2}) \\ \end{array} \right. .$

(2)

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$E'(u', v') = I(u', v'){\left( {\frac{2}{{1 + {{u'}^2} + {{v'}^2}}}} \right)^2} .$

(3)

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$\iint_\Omega {E(u, v)}{\rm{d}}u{\rm{d}}v = \iint_{\Omega '} {E'(u', v')}{\rm{d}}u'{\rm{d}}v' .$

(4)

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$E(u, v) = E'(u', v')\left| {\frac{{\partial u'}}{{\partial u}}\frac{{\partial v'}}{{\partial v}} - \frac{{\partial u'}}{{\partial v}}\frac{{\partial v'}}{{\partial u}}} \right| .$

(5)

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$\iint_\Omega {E(u, v)}{\rm{d}}u{\rm{d}}v = \iint_\Sigma {L(x, y)}{\rm{d}}\sigma $

()

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$ = \iint_\Sigma {L(x, y)}\sqrt {1 + {{(\frac{{\partial z}}{{\partial x}})}^2} + {{(\frac{{\partial z}}{{\partial y}})}^2}} {\rm{d}}x{\rm{d}}y, $

(6)

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$L(x, y)\sqrt {1 + {{(\frac{{\partial z}}{{\partial x}})}^2} + {{(\frac{{\partial z}}{{\partial y}})}^2}} \left| {\frac{{\partial x}}{{\partial u}}\frac{{\partial y}}{{\partial v}} - \frac{{\partial x}}{{\partial v}}\frac{{\partial y}}{{\partial u}}} \right| = E(u, v) .$

(7)

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$\left\{ \begin{array}{l} \frac{{\partial w}}{{\partial s}} = - \frac{{x - s}}{{z - w}} \\ \frac{{\partial w}}{{\partial t}} = - \frac{{y - t}}{{z - w}} \\ \end{array} \right. .$

(8)

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$\left\{ \begin{array}{l} \frac{{\partial w}}{{\partial u}} = \frac{{\partial w}}{{\partial s}}\frac{{\partial s}}{{\partial u}} + \frac{{\partial w}}{{\partial t}}\frac{{\partial t}}{{\partial u}} \\ \frac{{\partial w}}{{\partial v}} = \frac{{\partial w}}{{\partial s}}\frac{{\partial s}}{{\partial v}} + \frac{{\partial w}}{{\partial t}}\frac{{\partial t}}{{\partial v}} \\ \end{array} \right. .$

(9)

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$\left\{ \begin{array}{l} x = s + (z - w)\frac{{\frac{{\partial t}}{{\partial u}}\frac{{\partial w}}{{\partial v}} - \frac{{\partial t}}{{\partial v}}\frac{{\partial w}}{{\partial u}}}}{{\frac{{\partial s}}{{\partial u}}\frac{{\partial t}}{{\partial v}} - \frac{{\partial s}}{{\partial v}}\frac{{\partial t}}{{\partial u}}}} \\ y = t + (z - w)\frac{{\frac{{\partial s}}{{\partial v}}\frac{{\partial w}}{{\partial u}} - \frac{{\partial s}}{{\partial u}}\frac{{\partial w}}{{\partial v}}}}{{\frac{{\partial s}}{{\partial u}}\frac{{\partial t}}{{\partial v}} - \frac{{\partial s}}{{\partial v}}\frac{{\partial t}}{{\partial u}}}} \\ \end{array} \right. .$

(10)

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$\frac{{{\partial ^2}w}}{{\partial {u^2}}}\frac{{{\partial ^2}w}}{{\partial {v^2}}} - {\left( {\frac{{{\partial ^2}w}}{{\partial u\partial v}}} \right)^2} + {A_1}\frac{{{\partial ^2}w}}{{\partial {u^2}}}$

()

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$ + {A_2}\frac{{{\partial ^2}w}}{{\partial u\partial v}} + {A_3}\frac{{{\partial ^2}w}}{{\partial {v^2}}} + {A_4} = 0, $

(11)

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$ z = 100 + 12\left( {1 - {{(\frac{x}{{40}})}^2}} \right)\exp \left[ { - {{(\frac{x}{{40}})}^2} - {{(\frac{y}{{40}} + 1)}^2}} \right] \\ \;\;\; - 40\left( {\frac{x}{{200}} - {{(\frac{x}{{40}})}^3} - {{(\frac{y}{{40}})}^5}} \right)\exp \left[ { - {{(\frac{x}{{40}})}^2} - {{(\frac{y}{{40}})}^2}} \right] \\ $

()

Zexin Feng, Dewen Cheng, Yongtian Wang. Iterative freeform lens design for prescribed irradiance on curved target[J]. Opto-Electronic Advances, 2020, 3(7): 200010-1

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