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 typically tend to reduce the sharpness of the PSF. 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 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, and others, which makes the Zernike coefficients an intuitive framework to work with. It is 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, we are sometimes 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.

Analytical Forward Model

References

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Appendix

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