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

Shaping the Far Field of an Incident Light Beam

OpticalElement (OE)

Shaping the Far Field of an Incident Light Beam

OpticalElement (OE)

Shaping the Far Field of an Incident Light Beam

OpticalElement (OE)

Shaping the Far Field of an Incident Light Beam

OpticalElement (OE)

Shaping the Far Field of an Incident Light Beam

OpticalElement (OE)

Shaping the Far Field of an Incident Light Beam

OpticalElement (OE)

Shaping the Far Field: Inverse Argument

OpticalElement (OE)

Define a signal field in the far field with the

requested irradiance!

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!

Shaping the Far Field: Inverse Argument

OpticalElement (OE)

Shaping the Far Field: Inverse Argument

Shaping the Far Field: Inverse Argument

Phase is most essential design freedom.

Follows directly from a mapping of the signal field!

Inverse Method: Mapping

Inverse Method: Mapping

Inverse Method: Mapping

Shaping the Far Field: Inverse Argument

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• …

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.

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

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

Shaping the Far Field: Fourier Pair Synthesis

This design task has been tackled in the context of freeform design!

Shaping the Far Field: Fourier Pair Synthesis

Shaping the Far Field: Fourier Pair Synthesis

How does IFTA and “ray tracing” design methods fit together?

Far Field Shaping by Smooth Freeform Surface

Assume homeomorphic Fourier transform is numerically valid.

Far Field Shaping by Smooth Freeform Surface

Far Field Shaping by Smooth Freeform Surface

Far Field Shaping by Smooth Freeform Surface

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When is homeomorphic Fourier transform numerically justified?

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

Design Task in Math

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Source

Optical element

Target plane

wavefront phase response function

Design Task in Math

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Source

Optical element

Target plane

Design Task in Math

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Source

Optical element

Target plane

How to find the mapping ?

Design Task in Math

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Source

Optical element

Target plane

How to find the mapping ?

Source

Optical element

Target plane

Prepare the Fourier Pair from Given Informaiton

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Prepare the Fourier Pair from Given Informaiton

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irradiance

Source

Optical element

Target plane

Source

Optical element

Target plane

Design Homeomorphism between 𝝆𝝆 and 𝜿𝜿

Parseval's equation:

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Searching a mapping function:

Mathematical Model: Optimal Mass Transport Problem

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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

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

Mathematical Model: Optimal Mass Transport Problem

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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

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

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

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

From the Mapping to Phase

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Q: How to do the integration?

A: B-spline model

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

Simulation with the Functional Embodiment

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Source

Optical element

Target plane

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

Source

Optical element

Target plane

Task with Small Size Pattern

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150mm150m

m

Target pattern

decrease the size of the target pattern from 1m to 150mm

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

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

Iterative Fourier Transform Algorithm (IFTA) Optimization

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Wrapping

IFTA

Source

Optical element

Target plane

Comparison of the Result

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before optimization

irradiance

after optimization

irradiance

Source

Optical element

Target plane

Structure Design

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• How to design a structure to realize the functionality?

• Thin element approximation (TEA)?

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

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

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

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!

Thank You!