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.

More explicitly, if the phase due to aberrations is expressed in normalized Cartesian basis as: Failed to parse (unknown function "\math"): {\displaystyle e^{i\Phi(\kappa_x, \kappa_y)} = e^{i\Phi(\rho, \theta)} <\math>, the Zernike polynomial expansion of the aberration phase <math>\Phi} is given by:

\Phi(\rho, \theta) = \Sigma_{n, m} \alpha^m_n R^m_n(\rho) cos(m\theta)

where $0\leq \rho \leq 1,0\leq \theta \leq 2\pi$

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 as a function of the NA-normalized pupil coordinates (in Cartesian basis) as $\Phi (\kappa _{x},\kappa _{y})$, the point spread function can be expressed as:

$U(x,y;f)={\frac {1}{\pi }}\int _{\kappa _{x}^{2}+\kappa _{y}^{2}\leq 1}\int e^{if(\kappa _{x}^{2}+\kappa _{y}^{2})}e^{i\Phi (\kappa _{x},\kappa _{y})}e^{[2\pi i\kappa _{x}x+2\pi i\kappa _{y}y]}d\kappa _{x}d\kappa _{y}$

Note that this formula neglects the Radiometric correction factor, which becomes relevant for high numerical aperture systems. It also approximates the defocus phase for the small NA case. However, these approximations typically suffice for

We do not expect to be able to retrieve a complex phase from the single point spread function, unless we make additional prior assumptions regarding the phase. Instead, we use the approach proposed by Van Der Avoort in [2], which relies on using multiple point spread functions. In this approach, we hope to acquire a focal progression of point spread functions: $U(x,y,-f_{max})...,U(x,y,-deltaf),U(x,y,0),U(x,y,\delta f),...U(x,y,f_{max})$ and perform computations that will let us recover the zernike coefficients for the phase function

Analytical Forward Model

References

[1] M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, Cambridge, 2002)

[2] C. Van Der Avoort, J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, "Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer-Zernike approach", J. Mod. Opt., 52, 12, 2005, pp. 1695-1728

[3] A. J. E. M. Janssen, "Extended Nijboer-Zernike approach for the computation of optical point-spread functions", J. Opt. Soc. Am. A, 19, 5, 2002

[4] J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, A. S. van des Nes, "Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system", J. Opt. Soc. Am. A, 20, 12, 2003

[5] A. J. E. M. Janssen, J. J. M. Braat, P. Dirksen, "On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus", J. Mod. Opt., 51, 5, 2004, pp. 687-703

[6] P. Dirksen, J. Braat, A. J. E. M. Janssen, "Estimating resist parameters in optical lithography using the extended Nijboer-Zernike theory", J. Microlith., Microfab., Microsyst. 5(1), 013005, 2006

[7] Extended Nijboer-Zernike (ENZ) Analysis and Aberration Retrieval research site (Netherlands group): https://www.nijboerzernike.nl/

Appendix

The computational implementation for this project can be found on this iPython notebook: Code_okkeun.zip