Indrasen Bhattacharya

Introduction

Aberrations in the wavefront profile degrade the resolving power of imaging systems. It is highly desirable to measure and compensate any aberrations present in an imaging system. Such imaging systems may include microscopes, telescopes, lithography steppers, iPhone cameras, AR/VR goggles as well as the human eye. We shall consider the more traditional case of optical microscopes and lithography steppers in this work, but we do not rule out extensions to other imaging systems. The point spread function (PSF) at the image plane determines the resolving power of microscopes and the feature size in lithography systems. It is important to have a narrow PSF that is preferably diffraction limited by the numerical aperture (NA) of the imaging system. Wavefront aberrations reduce the sharpness of the PSF and lead to undesirable asymmetries and artifacts. The wavefront aberrations are typically expressed in an orthogonal radial basis on the unit circular pupil called the Zernike polynomials. The Zernike polynomials, together with azimuthal sinusoids, form a complete basis on the unit circle: any 2-dimensional phase function can be expressed as an appropriate linear combination of these polynomials. The weighting factors of the Zernike polynomials are termed as Zernike coefficients. Since any wavefront aberration profile can be fitted to a series of Zernike coefficients: the pupil function and Zernike coefficients are equivalent representations. Conveniently, the Zernike polynomials can be interpreted as specific physical aberrations such as spherical aberration, coma, astigmatism, trefoil and others, which makes the Zernike coefficients an intuitive framework to work with.

It is conceptually straightforward to calculate the point spread function from the wavefront profile: they are related by a Fourier transform where the aberrations occur in the phase of the pupil function. However, in this case we are interested in determining what is wrong with the system based on how the PSF appears. This is the inverse problem: determining the Zernike coefficients from the aberrated point spread function. It is important to explicitly state the inputs and outputs of the computational procedure we are hoping to undertake. If the wavefront aberration is given by $\Phi (\rho ,\theta )$, the point spread function can be expressed as:

$x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\hat {x}}_{1}(\omega )e^{it\omega }\,d\omega$

Analytical Forward Model

References

[1]

[2] Westheimer, G. (1960). Modulation thresholds for sinusoidal light distributions on the retina. Journal of Physiology, 152, 67-74

[3] http://www.ssc.education.ed.ac.uk/resources/vi&multi/VIvideo/cntrstsn.html

[4] Jennings, J. A., & Charman, W. N. (1981). Off-axis image quality in the human eye. Vision Research, 21(4), 445–455.

[5] Nicolas P. Cottaris, Brian A. Wandell, Fred Rieke, David H. Brainard; A computational observer model of spatial contrast sensitivity: Effects of photocurrent encoding, fixational eye movements, and inference engine. Journal of Vision 2020;20(7):17

[6] https://foundationsofvision.stanford.edu/chapter-7-pattern-sensitivity/

[7] http://usd-apps.usd.edu/coglab/CSFIntro.html

[8] Robson JG, Graham N. Probability summation and regional variation in contrast sensitivity across the visual field. Vision Res. 1981;21:409–18.

[9] Skalicky S.E. (2016) Contrast Sensitivity. In: Ocular and Visual Physiology. Springer, Singapore. https://doi.org/10.1007/978-981-287-846-5_20.

[10] https://github.com/isetbio/isetbio

[11] http://isetbio.org/

[12] Cortes, Corinna; Vapnik, Vladimir N. (1995). "Support-vector networks". Machine Learning. 20 (3): 273–297.

Appendix

The codes for this project can be found on this notebook: Code_okkeun.zip