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