From the iterative Fourier transform algorithm (IFTA) to ... · A Least -Squares Method for Optimal...
Transcript of From the iterative Fourier transform algorithm (IFTA) to ... · A Least -Squares Method for Optimal...
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From the iterative Fourier transform algorithm (IFTA) to “ray mapping” and back
L. Yang*, R. Knoth**, I. Badar*, C. Hellmann***, F. Wyrowski**University of Jena, ** LightTrans GmbH, ***Wyrowski Photonics
EOS Topical Meeting on Diffractive Optics, September 2019, Jena, Germany
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Shaping the Far Field of an Incident Light Beam
OpticalElement (OE)
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Shaping the Far Field of an Incident Light Beam
OpticalElement (OE)
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Shaping the Far Field of an Incident Light Beam
OpticalElement (OE)
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Shaping the Far Field of an Incident Light Beam
OpticalElement (OE)
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Shaping the Far Field of an Incident Light Beam
OpticalElement (OE)
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Shaping the Far Field of an Incident Light Beam
OpticalElement (OE)
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Shaping the Far Field: Inverse Argument
OpticalElement (OE)
Define a signal field in the far field with the
requested irradiance!
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Shaping the Far Field: Inverse Argument
OpticalElement (OE)
Define a signal field in the far field with the
requested irradiance!
Phase is a design freedom!
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Shaping the Far Field: Inverse Argument
OpticalElement (OE)
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Shaping the Far Field: Inverse Argument
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Shaping the Far Field: Inverse Argument
Phase is most essential design freedom.
Follows directly from a mapping of the signal field!
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Inverse Method: Mapping
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Inverse Method: Mapping
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Inverse Method: Mapping
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Shaping the Far Field: Inverse Argument
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Shaping the Far Field: Type of Element
• Smooth height profile (one or more maxima)• Stepped profile (one or more maxima)• Zone-type profile (smooth with jumps)• Meta-structured layer • GRIN layer• Combinations of different profile types• …
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Shaping the Far Field: Fourier Pair Synthesis
• Smooth height profile (one or more maxima)• Stepped profile (one or more maxima)• Zone-type profile (smooth with jumps)• Meta-structured layer • GRIN layer• Combinations of different profile types• … Field follows from structure of
optical element and fabrication constraints.
Fourier pair synthesis: Find an OE which generates a field which fits with the constraint in Fourier domain.
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Shaping the Far Field: Fourier Pair Synthesis
• Smooth height profile (one or more maxima)• Stepped profile (one or more maxima)• Zone-type profile (smooth with jumps)• Meta-structured layer • GRIN layer• Combinations of different profile types• … Field follows from structure of
optical element and fabrication constraints.
Fourier pair synthesis: Find an OE which generates a field which fits with the constraint in Fourier domain.
Fourier pair synthesis by parametric optimization
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Shaping the Far Field: Fourier Pair Synthesis
• Smooth height profile (one or more maxima)• Stepped profile (one or more maxima)• Zone-type profile (smooth with jumps)• Meta-structured layer • GRIN layer• Combinations of different profile types• … Field follows from structure of
optical element and fabrication constraints.
Fourier pair synthesis: Find an OE which generates a field which fits with the constraint in Fourier domain.
IFTA: Iterative Fourier Transform Algorithm
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Shaping the Far Field: Fourier Pair Synthesis
This design task has been tackled in the context of freeform design!
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Shaping the Far Field: Fourier Pair Synthesis
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Shaping the Far Field: Fourier Pair Synthesis
How does IFTA and “ray tracing” design methods fit together?
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Far Field Shaping by Smooth Freeform Surface
Assume homeomorphic Fourier transform is numerically valid.
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Far Field Shaping by Smooth Freeform Surface
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Far Field Shaping by Smooth Freeform Surface
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Far Field Shaping by Smooth Freeform Surface
?
When is homeomorphic Fourier transform numerically justified?
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Design Task
1m
1m
SourceGaussian wave: 532nmFull divergent angle: 6°
Optical elementSize: 1 × 1mm
1m
Target planeTarget pattern
? How to design the optical element for achieving the target irradiance distribution on target plane?
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• large pattern size• long progation distance
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Design Task in Math
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Source
Optical element
Target plane
wavefront phase response function
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Design Task in Math
30
Source
Optical element
Target plane
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Design Task in Math
31
Source
Optical element
Target plane
How to find the mapping ?
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Design Task in Math
32
Source
Optical element
Target plane
How to find the mapping ?
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Source
Optical element
Target plane
Prepare the Fourier Pair from Given Informaiton
33
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Prepare the Fourier Pair from Given Informaiton
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irradiance
Source
Optical element
Target plane
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Source
Optical element
Target plane
Design Homeomorphism between 𝝆𝝆 and 𝜿𝜿
Parseval's equation:
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Searching a mapping function:
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Mathematical Model: Optimal Mass Transport Problem
36
Prins, C.; et al. A Least-Squares Method for Optimal Transport Using the Monge-Ampere Equation SIAM Journal on Scientific Computing, 2015, 37, B937-B961
Given two density functions and defined in two bounded supports: and , with equal masses of a given material
A mapping is searched to realize the transfer of density function from to , and minimize the cost function
If is a smooth one-to-one map, it leads to a local equation
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Mathematical Model: Optimal Mass Transport Problem
37
Prins, C.; et al. A Least-Squares Method for Optimal Transport Using the Monge-Ampere Equation SIAM Journal on Scientific Computing, 2015, 37, B937-B961
Given two density functions and defined in two bounded supports: and , with equal masses of a given material
A mapping is searched to realize the transfer of density function from to , and minimize the cost function
If is a smooth one-to-one map, it leads to a local equation
The physical quantity of our case and are assigned to the equation
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Optimal Mass Transport Algorithm
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Prins, C.; et al. A Least-Squares Method for Optimal Transport Using the Monge-Ampere Equation SIAM Journal on Scientific Computing, 2015, 37, B937-B961
The meshes show the bijective map, which is essential property for
the HFT
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From the Mapping to Phase
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Q: How to do the integration?
• We assume the phase function is represented by B-spline functions:
A: B-spline model
• Take gradient:
• Fit the gradient functions of the model with the data set simultaneously and obtain the control point
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From the Mapping to Phase
40
Q: How to do the integration?
A: B-spline model
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From the Mapping to Phase
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Q: How to do the integration?
A: B-spline model
• Modelling: wavefront phase homeomorphic Fourier transform
• Inverse design: homeomorphism wavefront phase
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Simulation with the Functional Embodiment
42
Source
Optical element
Target plane
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Source
Optical element
Target plane
Simulation with the Functional Embodiment
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Source
Optical element
Target plane
Simulation with the Functional Embodiment
simulation with HFT simulation with rigorous Fourier transform
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irradiance irradiance
Validity of homeomorphism assumption is proofed
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Source
Optical element
Target plane
Task with Small Size Pattern
45
150mm150m
m
Target pattern
decrease the size of the target pattern from 1m to 150mm
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Simulation with the Functional Embodiment
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simulation with HFT
irradiance
simulation with rigorous Fourier transform
irradiance
• diffraction effect appears obviously• homeomorphism assumption fails
Source
Optical element
Target plane
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Iterative Fourier Transform Algorithm (IFTA) Optimization
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Wrapping
Assuming homeomorphism can be used to generate vortex-free initial distribution
for further optimization.
Source
Optical element
Target plane
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Iterative Fourier Transform Algorithm (IFTA) Optimization
48
Wrapping
IFTA
Source
Optical element
Target plane
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Comparison of the Result
49
before optimization
irradiance
after optimization
irradiance
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Source
Optical element
Target plane
Structure Design
50
• How to design a structure to realize the functionality?
• Thin element approximation (TEA)?
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Source
Optical element
Target plane
Freeform Surface Design
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Algorithm in brief:1. Initializing a reference plane 2. Input and output wave vector: 𝒌𝒌in 𝑥𝑥,𝑦𝑦 ,𝒌𝒌out 𝑥𝑥,𝑦𝑦
3. Gradient of the surface height 𝛻𝛻𝐻𝐻(𝑥𝑥,𝑦𝑦)
4. Surface height with B-spline model 𝐻𝐻 𝑥𝑥,𝑦𝑦
5. Update the plane of step 1 with 𝐻𝐻 𝑥𝑥,𝑦𝑦 and iteratively perform step 1-4
Yang, L.; Knoth, R.; Hellmann, C. & Wyrowski, Proc. SPIE, 2018, 10518
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Freeform Surface Design
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Source
Target planeFreeform component
3D View 2D Profile
• Freeform component with predefine planar surface
• Freeform surface represented by B-spline functions
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Freeform Surface Design
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3D View 2D Profile
Field tracing result: Irradiance
• The designed gradient of the surface doesn‘t guarantee integrable
• B-spline model obtains an optimal approximation of the surface
Source
Target planeFreeform component
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Conclusion
• Shaping the far field of an incident field requires mainly a Fourier pair synthesis.
• Selection of the type of component and related fabrication technology determines the field constraints in the space domain.
• Selection of the signal in the target determines constraint in Fourier domain. • Dependent on the design situation the Fourier transform can be:
− Homeomorphic − Non-homeomorphic
• That results accordingly to different design algorithms:− Mapping design algorithm between x- and k-domain− IFTA
• IFTA can often benefit from an initial design by the mapping algorithm.
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Conclusion
• Shaping the far field of an incident field requires mainly a Fourier pair synthesis.
• Selection of the type of component and related fabrication technology determine the field constraints in the space domain.
• Selection of the signal in the target determine constraint in Fourier domain. • Dependent on the design situation the Fourier transform can be:
− Homeomorphic − Non-homeomorphic
• That results accordingly to different design algorithms:− Mapping synthesis algorithm into k-domain− IFTA
• IFTA can often benefit from an initial design by the mapping algorithm.
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The concept of homeomorphic operations in physical-optics modeling and design provides newinsights and options in light shaping!
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