1 CASA-day April 6, 2016
A new ray tracing method
based on phase space
Carmela Filosa
Jan ten Thije Boonkkamp
Wilbert IJzerman
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Phase space based ray tracing
โข Introduction of ray tracing
โ A two-dimensional optical system: the TIR-collimator
โ Physical background
โ Issues with ray tracing
โข The idea: phase space based ray tracing
โ How to improve ray tracing
โ Boundaries are mapped onto boundaries
โข Implementation
โ Refinement procedure
โ Boundary detection
โ Calculation of intensity
โข Results
โ Speed of convergence
โข Conclusion and outlook
Outline
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Ray tracing is like playing billiards
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Ray tracing for a TIR-collimator
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Background
๐ 2 =๐1๐2๐ 1 + 1 โ
๐1๐2
2
+๐1๐2
2
๐, ๐ 1 2 โ๐1๐2๐, ๐ 1 ๐ ๐
๐ก2 = ๐ก1 โ 2 ๐ก1, ๐ ๐
Law of reflection::
Law of refraction::
I = ๐ฯ
๐๐ก ๐๐ = ๐๐/๐๐๐
L = ๐2ฯ
cos(๐ก) ๐๐๐๐ก ๐๐/๐
Intensity::
Luminance::
รtendue:: ๐๐ธ = ๐ ๐๐ cos(๐ก) ๐๐ก ๐ ๐๐๐
Flux:: Measure of the perceived power of light by the human eye, (ฯ = ๐๐ ) .
n reflected ray incident ray
โข Shoot rays from source to target randomly
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What is ray tracing?
Our model
โข Intensity calculated by counting the rays in bins
Convergence (Compared to a simulation with a large number of rays)
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โข Full system is characterized by phase space
โข ๐ซ๐ : ๐ฎ๐ โ1,1 (position ๐ฅโ๐ฎ and angle ฯ = sin ๐ก) ๐ซ๐ก: ๐ฏ๐ โ1,1 (position ๐โ๐ฏ and angle ฮท = sin ฮธ).
โข The optical map โณ:๐ซ๐ ๐ซ๐ก maps region to region and
boundary to boundary (Riesz)
How to improve ray tracing?
๐ซ๐ ๐ซ๐ก
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Implementation: How to find the boundaries on source phase space?
โข On source phase space: take a coarse grid
โข If paths are different refine unless already fine enough
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Implementation: How to find the boundaries on target phase space?
Method 1: use alpha-shapes 1
For a cloud of points, like eating stracciatella ice-cream with a spoon without eating the chocolate
1 F. Bernardini and C. L. Bajaj, โSampling and reconstructing manifolds using alpha-shapes,โ (1997)
Issue: how to choose the size of your spoon? (We used an รtendue argument)
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Method 2: use that boundary is mapped on boundary for regions of phase space 1
Implementation: How to find the boundaries on target phase space?
1 H. Ries and A. Rabl, โEdge-ray principle of nonimaging optics,โ JOSA A 11, 2627โ2632 (1994)
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Implementation: How to find the boundaries on target phase space?
1 H. Ries and A. Rabl, โEdge-ray principle of nonimaging optics,โ JOSA A 11, 2627โ2632 (1994)
Method 2: use that boundary is mapped on boundary for regions of phase space 1
More points around the boundaries
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Implementation: How to find the boundaries on target phase space?
To calculate intensity: integrate along a line If we assume constant brightness (Lambertian source) We obtain:
๐ผ๐๐ ฮท = ๐ฟ๐ก ๐, ฮท ๐๐๐ฮ ๐,2๐ ฮท
๐ฮ ๐,2๐โ1 ฮท๐,๐
๐ฟ๐ ๐ฅ, ๐ = ๐ฟ๐ก ๐, ฮท = 1
๐ผ๐๐ ฮท = ๐ฮ ๐,2๐ ฮท โ ๐ฮ ๐,2๐โ1 ฮท
๐,๐
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โข Method calculates intensity correctly
Results
โข Convergence is 1/N instead of 1/ ๐
PS intensity with around 105 rays.
Reference intensity: MC ray tracing with 108 rays.
Using ray tracing on PS far fewer rays need to be traced to obtain the same accuracy.
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Conclusions and outlook
โข This method is 2D only (meridional plane) but can be extended to 3D as well
โข More rapid convergence 1/N versus 1/ ๐
โข Future work: include Fresnel reflection and scattering phenomena
Acknowledgement:
I would like to thank Jorg Portegies for helpful discussions and providing his master thesis on which this work is based.
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How to choose the parameter ฮฑ for the ฮฑ-shapes method
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Stopping criterion for the triangulation refinement