About the Optimal Optical Transfer Function of a Microscope Jörg Enderlein Georg August University Göttingen Germany Quantitative BioImaging 2019, January 9 ‒ 11 Rennes, France
The Optical Microscope And Its Resolution y min 2 sin n 2 N.A.
The Optical Transfer Function detection and imaging as a superposition of plane waves Abbe‘s sine condition: M sin ' n sin
The Optical Transfer Function imaging as a superposition of plane waves Fourier transform of electric field
The Optical Transfer Function intensity = absolute square of electric field in Fourier space: convolution Fourier transform of electric field
The Optical Transfer Function Fourier transform of Optical Transfer electric field Function
The Optical Transfer Function PSF = Fourier back- Optical Transfer transform of OTF Function
The Point Spread Function electric field intensity
The Optical Transfer Function electric field Fourier transform of electric field
The Optical Transfer Function convolution Fourier transform of electric field
The Optical Transfer Function convolution OTF
The Optical Transfer Function electric field of collected light OTF WFM or or electric field of focused excitation OTF CLSM (no pinhole)
The Optical Transfer Function 4 n 2 n max k sin max k sin 2 y min 2 sin n 2 N.A. max k
But what would be the optimal Optical Transfer Function for a given finite frequency support of a microscope? Answer: The OTF which yields the sharpest image.
OTF One-dimensional case q max dq lim exp iqx x 2 q max q max Maybe, a uniform amplitude PSF across all available frequencies is optimal. But we have also to require strict non-negativeness of PSF!
OTF One-dimensional case 2 q max dq U x exp iqx 2 q max PSF strictly non-negative
OTF E-Field How about non-uniform amplitude distribution? PSF q a q 1
A heuristic argument for the Optimal Optical Transfer Function What is a good measure of „ sharpness “? 2 dxU x x dxU x does generally not converge So what about the following idea? 2 2 2 dxU x x dxU x minimize
A heuristic argument for the Optimal Optical Transfer Function 2 2 2 dxU x x dxU x q a q 1
Recipe For Finding Optimal OTF: Minkowski Sum Decomposition A A B Given the frequency support of OTF (B), search Minkowski decomposition (A) so that its auto-convolution recovers the OTF support, then fill Minkowksi decomposition with uniform frequency amplitude and calculate via auto- convolution optimal OTF.
OTFs of Ideal Wide-Field Microscope or Pinhole-less Laser-Scanning Microscope Are Optimal electric field of collected light OTF WFM or or electric field of focused excitation OTF CLSM (no pinhole)
Minkowski Sum = Geometric Convolution A B
Minkowski Sum = Geometric Convolution A B
Minkowski Sum = Geometric Convolution C A B
Minkowski Sum Decomposition easy for convex shapes generally unsolved problem
Example: Ideal Confocal Microscope Or Image Scanning Microscope OTF of excitation OTF of detection OTF of LSCM through infinitely with infinitely small pinhole small pinhole Stokes shift neglected
Example: Ideal Confocal Microscope
Example: Ideal Confocal Microscope
Conclusion • Optimal OTF can be generated via an auto- convolution of a frequency distribution with uniform constant amplitude • General problem of how to find Minkowski decomposition for non-convex shapes - may be complicated • Knowing the optimal OTF is important for deconvolution but also for physical PSF engineering
Acknowledgements Ingo Gregor Alexey I. Chizhik Anna M. Chizhik Narain Karedla Roman Tsukanov Yunking Tang Simon Stein Jan Thiart Daja Ruhlandt Collaborations Sebastian Isbaner Arindam Ghosh Fred Wouters Akshita Sharma Gertrude Bunt Aditya Katta Jörg Großhans Soheil Mojiri Robert Ros Shun Qin Andreas Janshoff Hongje Jang Florian Rehfeldt Nazar Oleksiievits Ralph Kehlenbach www.joerg-enderlein.de
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