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Efficient wave-optical calculation of 'bad systems'
1
Norman G. Worku,2 Prof. Herbert Gross1,2
25.11.2016
(1) Fraunhofer Institute for Applied Optics and Precision Engineering IOF, Jena, Germany(2) Institute of Applied Physics, Friedrich-Schiller-University Jena, Germany
Motivation
PSF computation for
„bad systems“ –
• high wave aberration
on non-planar image surface
• curved image sensors for smallsmartphone camers
Commonly used hybrid diffraction model (ray tracing + diffraction propagation)
well defined and undistorted exit pupil
image plane perpendicular to the light cone.
field at arbitrary point can not be determined
2
Source: U.S. Patent No. 9,244,253.
Source: Wikipedia.
Source: Gross, H.,
Outline
Introduction
Complex ray tracing
Single Gaussian beam
• Example 1: Propagation to curved plane
• Example 2: Propagation through non-orthogonal system
Gaussian decomposition
• Example 3: PSF of aberrated system
Conclusion
3
4Introduction
Gaussian beam decomposition method
Step 1: Decomposition
• Generate set of Gaussian beams at input aperture
Step 2: Propagation
• Each Gaussian beam is propagated totarget plane.
Step 3: Coherent superposition
• OPL of central ray is added as phasefactor on each beam.
• Point wise addition of the complexfield contributions from each beam.
Source: Greynolds, Alan W., SPIE 2014.
Trace generally astigmatic Gaussian beam through optical systemsusing ray tracing [1].
Gaussian beam representation: set of 5 rays
5Complex ray tracing
Divergence and waist rayparameters a complex rayparameters Real part : divergence ray
parameters Imaginary part : waist ray
parameters
Two rays with complex parameters: „complex rays“• ℎ1, ℎ2, 𝑢1 𝑎𝑛𝑑 𝑢2: position and direction vectors of each complex ray.
• Condition: ℎ1. 𝑢2 − ℎ2. 𝑢1 = 0
[1]. Arnaud, J.A., Applied Optics, 1985.
Gaussian field from set of complex rays
𝐸 𝑟 =𝐸0
ℎ1 × ℎ2
𝑒𝑖𝑘
ℎ1× 𝑟 𝑢2 . 𝑟 − ℎ2× 𝑟 𝑢1 . 𝑟
2ℎ1×ℎ2
Where ℎ1, ℎ2, 𝑢1 𝑎𝑛𝑑 𝑢2: the complex ray parameters.
6Complex ray tracing…
Source: R. Wilhelm, B. Koehler et al, 2002.
Amplitude and phase profiles of a single Gaussian beam on curvedsurface with small curvatures
7Example 1: Propagation to curved surface
𝐶𝑥 = 𝐶𝑦 = 2 ∗ 10−4 𝑚𝑚
Gaussian beam, waist radius = 2mm
Slightly curved surfaceProp. dist = 10 mm
𝐶𝑥 = 𝐶𝑦 = 0 𝐶𝑥 = 1 ∗ 10−3 𝑚𝑚,𝐶𝑦 = 0
Rotated cylindrical mirror
Single cylindrical focusing mirror, with f = 100 mm,
Oriented at 45° relative to the axes of the input Gaussian beam axis.
Gaussian beam width of 2 mm and 1 mm in x and y respectively.
8Example 2: Non-orthogonal system
Amplitude profiles of the Gaussian (λ =1 𝜇𝑚) after the mirror compared with result from the original paper [2]
Amplitude profile rotates in free space generally astigmatic Gaussian beam
For larger wavelength of λ = 10 𝜇𝑚,
9Rotated cylindrical mirror …
[2]. Greynolds, Alan W., International Society for Optics and Photonics, 1986.
Large diffraction effects Large beam width at the focal plane Rotation of ellipse at focal plane != 45
deg
Aberrated optical system
Single freeform focusing mirror, with R = -200 mm, conic = -1 and zernikefringe sag terms (Z9 – Spherical , Z8 - Coma, Z6 - Astigmatism ) and λ = 0.5 µm.
Input: plane wave through circular aperture of diameter = 200 mm placed at front focal plane.
10Example 3: PSF with large aberration
Gaussian decomposition of input beam - Grid of 41 X 41 Gaussian beams- Overlap factor of 1.5
11PSF with large aberration …
PSF without aberration compared with Zemaxresult for validation.
Spherical Aberration Coma Astigmatism
12PSF with large aberration …
Intensity around focal plane in the presence of large aberration - coma.
Peak of the intensity profile: moves on a curve for different z planes (bananicity). shifted in the transversal due to the tilt of the wavefront – shift in chief ray
position.
Computational effort ( for single mirror system)
Number of Gaussian beam 32X32 32X32 64X64 32X32
Grid size of field evaluation 32X32 128X128 128X128 256X256
Total computation time (sec) 0.736 3.455 9.443 16.848
Field propagation using Gaussian decomposition
Provides end-to-end mechanism for propagating fields through opticalsystems.
Can be used for systems with high aberration.
Complex field value can be computed at any point independantly.
Limitations
Each beam should be locally paraxial
Sharp edge apertures – smooth Gaussian edge
Outlook
Decomposition of arbitrary input fields
13Concluding remarks
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