Efficient wave-optical calculation of 'bad systems'

Transcription

1 1 Efficient wave-optical calculation of 'bad systems' Norman G. Worku, 2 Prof. Herbert Gross 1, (1) Fraunhofer Institute for Applied Optics and Precision Engineering IOF, Jena, Germany (2) Institute of Applied Physics, Friedrich-Schiller-University Jena, Germany

2 Motivation 2 PSF computation for bad systems high wave aberration on non-planar image surface Source: Wikipedia. curved image sensors for small smartphone camers Commonly used hybrid diffraction model (ray tracing + diffraction propagation) Source: Gross, H., well defined and undistorted exit pupil image plane perpendicular to the light cone. field at arbitrary point can not be determined Source: U.S. Patent No. 9,244,253.

3 Outline 3 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

4 Introduction 4 Gaussian beam decomposition method Step 1: Decomposition Generate set of Gaussian beams at input aperture Step 2: Propagation Each Gaussian beam is propagated to target plane. Step 3: Coherent superposition OPL of central ray is added as phase factor on each beam. Source: Greynolds, Alan W., SPIE Point wise addition of the complex field contributions from each beam.

5 Complex ray tracing 5 Trace generally astigmatic Gaussian beam through optical systems using ray tracing [1]. Gaussian beam representation: set of 5 rays Divergence and waist ray parameters a complex ray parameters Real part : divergence ray parameters Imaginary part : waist ray parameters Two rays with complex parameters: complex rays h 1, h 2, u 1 and u 2 : position and direction vectors of each complex ray. Condition: h 1. u 2 h 2. u 1 = 0 [1]. Arnaud, J.A., Applied Optics, 1985.

6 Complex ray tracing 6 Gaussian field from set of complex rays E r = E 0 h 1 h 2 e ik h 1 r u 2. r h 2 r u 1. r 2h 1 h 2 Where h 1, h 2, u 1 and u 2 : the complex ray parameters. Source: R. Wilhelm, B. Koehler et al, 2002.

7 Example 1: Propagation to curved surface 7 Amplitude and phase profiles of a single Gaussian beam on curved surface with small curvatures Gaussian beam, waist radius = 2mm Prop. dist = 10 mm Slightly curved surface Cx = Cy = 0 Cx = Cy = mm Cx = mm, Cy = 0

8 Example 2: Non-orthogonal system 8 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.

9 Rotated cylindrical mirror 9 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, Large diffraction effects Large beam width at the focal plane Rotation of ellipse at focal plane!= 45 deg [2]. Greynolds, Alan W., International Society for Optics and Photonics, 1986.

10 Example 3: PSF with large aberration 10 Aberrated optical system Single freeform focusing mirror, with R = -200 mm, conic = -1 and zernike fringe 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. Gaussian decomposition of input beam - Grid of 41 X 41 Gaussian beams - Overlap factor of 1.5

11 PSF with large aberration 11 PSF without aberration compared with Zemax result for validation. Spherical Aberration Coma Astigmatism

12 PSF with large aberration 12 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 128X X X256 Total computation time (sec)

13 Concluding remarks 13 Field propagation using Gaussian decomposition Provides end-to-end mechanism for propagating fields through optical systems. 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