Extended Nijboer-Zernike Analysis and Aberration Retrieval

Journal papers

3. P. Dirksen, J.J.M. Braat, A.J.E.M. Janssen, C.A.H. Juffermans, "Aberration retrieval using the extended Nijboer-Zernike approach," Journal of Microlithography, Microfabrication, and Microsystems 2 (2003), pp. 61-68.

4. J.J.M. Braat, P. Dirksen, A.J.E.M. Janssen, A. van de Nes, "Extended Nijboer-Zernike representation of the field in the focal region of an aberrated high-aperture optical system,"  J. Opt. Soc. Am. A 20 (2003), pp. 2281-2292; also in Virtual Journal of Biological Physics Research 6 (2003), issue 12.

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 (2004), pp. 687-703.

6. 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 (2005), pp. 1695-1728.

7. J.J.M. Braat, P. Dirksen, A.J.E.M. Janssen, A.S. van de Nes, S. van Haver "Extended Nijboer-Zernike approach to aberration and birefringence retrieval in a high-numerical-aperture optical system," J. Opt. Soc. Am. A. 22 (2005), pp. 2635-2650.

8. P. Dirksen, J.J.M. Braat, A.J.E.M. Janssen, "Estimating resist parameters using the Extended Nijboer-Zernike theory," Journal of Microlithography, Microfabrication, and Microsystems 5 (2006), 013005, pp. 1-11.

9. S. van Haver, J.J.M. Braat, P. Dirksen, A.J.E.M. Janssen, "High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory," J. Eur. Opt. Soc. Rapid Publ. 1 (2006), 06004, pp. 1-8.

10. A.J.E.M. Janssen, P. Dirksen, "Concise formula for the Zernike coefficients of scaled pupils," Journal of Microlithography, Microfabrication, and Microsystems 5 (2006), 030501, pp. 1-3.

11. A.J.E.M. Janssen, S. van Haver, J.J.M. Braat, P. Dirksen, "Strehl ratio and optimum focus of high-numerical-aperture beams," J. Eur. Opt. Soc. Rapid Publ. 2 (2007), 07008, pp. 1-9.

12. S. van Haver, J.J.M. Braat, P. Dirksen, A. J.E.M. Janssen, "High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory - Erratum," J. Eur. Opt. Soc. Rapid Publ. 2 (2007), 07011e, p. 1.

13. A.J.E.M. Janssen, P. Dirksen, "Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform," J. Eur. Opt. Soc. Rapid Publ. 2 (2007), 07012, pp. 1-3.

14. J.J.M. Braat, S. van Haver, A.J.E.M. Janssen, P. Dirksen, "Energy and momentum flux in a high-numerical-aperture beam using the extended Nijboer-Zernike diffraction formalism," J. Eur. Opt. Soc. Rapid Publ. 2 (2007), 07032, pp. 1-13.

15. A.J.E.M. Janssen, S. van Haver, P. Dirksen, J.J.M. Braat, "Zernike representation and Strehl ratio of optical systems with variable numerical aperture," J. Mod. Opt. 55 (2008), pp. 1127-1157.

16. R.M. Aarts, J.J.M. Braat, P. Dirksen, S. van Haver, C. van Heesch, A.J.E.M. Janssen, "Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture," J. Eur. Opt. Soc. Rapid Publ. 3 (2008), 08039, pp. 1-10.

17. S. van Haver, J.J.M. Braat, A.J.E.M. Janssen, O.T.A. Janssen, S.F. Pereira, "Vectorial aerial-image computations of three-dimensional objects based on the extended Nijboer-Zernike theory," J. Opt. Soc. Am. A 26 (2009), pp. 1221-1234.

18. J.J.M. Braat, S. van Haver, A.J.E.M. Janssen, S.F. Pereira, "Image formation in a multilayer using the Extended Nijboer-Zernike theory," J. Eur. Opt. Soc. Rapid Publ. 4 (2009), 09048, pp. 1-12.

19. H.P. Urbach, O.T.A. Janssen, S. van Haver, A.J.H. Wachters, "On the modeling of optical systems containing elements of different scales," J. Mod. Opt. iFirst (2010), pp. 1-13.

20. A.J.E.M. Janssen, "New analytic results for the Zernike circle polynomials from a basic result in the Nijboer-Zernike diffraction theory," J. Eur. Opt. Soc. Rapid Publ. 6 (2011), 11028, pp. 1-14.

21. A.J.E.M. Janssen, "Computation of Hopkins' 3-circle integrals using Zernike expansions," J. Eur. Opt. Soc. Rapid Publ. 6 (2011), 11059, pp. 1-6.

22. J.J.M. Braat, A.J.E.M. Janssen, "Double Zernike expansion of the optical aberration function from its power series expansion," J. Opt. Soc. Am. A 30 (2013), pp. 1213-1222.

23. S. van Haver, A.J.E.M. Janssen, "Advanced analytic treatment and efficient computation of the diffraction integrals in the Extended Nijboer-Zernike theory," J. Eur. Opt. Soc. Rapid Publ. 8 (2013), 13044, pp. 1-29.

24. A. P. Konijnenberg, L. Wei, N. Kumar, L. Couto Correa Pinto Filho, L. Cisotto, S. F. Pereira, H. P. Urbach, "Demonstration of an optimised focal field with long focal depth and high transmission obtained with the Extended Nijboer-Zernike theory," Opt. Express 22 (2014), pp. 311-324.

25. A.J.E.M. Janssen, "Zernike expansion of derivatives and Laplacians of the Zernike circle polynomials," J. Opt. Soc. Am. A 31 (2014), pp. 1604-1613.

26. S. van Haver, A.J.E.M. Janssen, "Truncation of the series expressions in the advanced ENZ-theory of diffraction integrals," J. Eur. Opt. Soc. Rapid Publ. 9 (2014), 14042, pp. 1-13.

27. J.J.M. Braat, A.J.E.M. Janssen, "Derivation of various transfer functions of ideal or aberrated imaging systems from the three-dimensional transfer function," J. Opt. Soc. Am. A 32 (2015), pp. 1146-1159.

28. R.S. Biesheuvel, A.J.E.M. Janssen, P. Pozzi, S.F. Pereira, "Implementation and benchmarking of a crosstalk-free method for wavefront Zernike coefficients reconstruction using Shack-Hartmann sensor data," OSA Continuum 1(2) (2018), pp. 581-603.

29. J.J.M. Braat, "The invention of the phase contrast microscope by Frits Zernike," NTvN 89(11) (2023), pp. 32-37 (English translation of the original publication in Dutch). Original publication: "De uitvinding van de fasecontrastmicroscoop door Frits Zernike."

30. Joseph J.M. Braat, "Analytic results for the partially coherent edge response of a perfect imaging system," Opt. Continuum 5, 651-673 (2026); https://doi.org/10.1364/OPTCON.575658.
The mauscript builds on Zernike's initial calculation (publication from 1934) of the central intensity in a knife-edge image for a 1D optical system with equal illumination and imaging aperture.

Book

1. Joseph Braat and Peter Torok, "Imaging Optics," ISBN 978-1-108-42808-8 (hardback, xii + 973 pages, 557 figures and 63 tables), Cambridge University Press, Cambridge, U.K. (2019).

Scalar and vector diffraction theory is the subject of Part III of this book. ENZ-theory in particular is treated in Chapter 9 (pages 609-656) and in Appendix F. Orthogonal polynomials on the unit disc (with emphasis on the Zernike polynomials) are discussed in Appendix D (pages 888-906).

Book chapters

1. J.J.M. Braat, S. van Haver, A.J.E.M. Janssen, P. Dirksen, "Assessment of optical systems by means of point-spread functions," in Progress in Optics, Vol. 51, E. Wolf, ed., (Elsevier, Amsterdam, The Netherlands, 2008), pp. 349-468.

Ph.D. Dissertations

1. A.S. van de Nes, "Rigorous Electromagnetic Field Calculations for Advanced Optical Systems," Ph. D. Dissertation Delft University of Technology (2005), pp. 1-164.

2. S. van Haver, "The Extended Nijboer-Zernike diffraction theory and its applications," Ph. D. Dissertation Delft University of Technology (2010), pp. 1-176.

Master thesis

1. S. van Haver, "Extended Nijboer-Zernike diffraction and aberration retrieval theory for high-numerical-aperture optical imaging systems," December 2005.

Conference Proceedings

1. P. Dirksen, J.J.M. Braat, P. De Bisschop, A.J.E.M. Janssen, C.A.H. Juffermans, Alvina Williams, "Characterization of a projection lens using the extended Nijboer-Zernike approach," Proc. SPIE 4691, Santa Clara, March 2002, pp. 1392-1399.

2. P. Dirksen, J. Braat, A.J.E.M. Janssen, C. Juffermans, A. Leeuwestein, "Experimental determination of lens aberrations from the intensity point-spread function in the focal region," Proc. SPIE 5040, Santa Clara, February 2003, pp. 1-10.

3. P. Dirksen, J. Braat, A.J.E.M. Janssen, A. Leeuwestein, H. Kwinten, D. Van Steenwinckel, "Determination of resist parameters using the extended Nijboer-Zernike theory," Proc. SPIE 5377, Santa Clara, February 2004, pp. 150-159.

4. P. Dirksen, J.J.M. Braat, A.J.E.M. Janssen, A. Leeuwestein, "Aberration retrieval for high-NA optical  systems using the extended Nijboer-Zernike theory," Proc. SPIE 5754, San Jose, USA, February-March 2005, pp. 262-273.

5. J.J.M. Braat, P. Dirksen, A.J.E.M. Janssen, "Through-focus point-spread function evaluation for lens metrology using the extended Nijboer-Zernike theory," in Fringe 2005 (W. Osten, ed.), Springer, Berlin, 2005, pp. 299-307. 

6. P. Dirksen, J.J.M. Braat, A.J.E.M. Janssen, A. Leeuwestein, T. Matsuyama, T. Noda, "Aerial image based lens metrology for wafer steppers," Proc. SPIE 6154, San Jose, February 2006, 61540X, pp. 1-11.

7. S. van Haver, A.J.E.M. Janssen, P. Dirksen, J.J.M. Braat, "Extended Nijboer-Zernike (ENZ) based evaluation of amplitude and phase aberrations on scaled and annular pupils," meeting digest EOS Advanced Imaging Techniques 2007, Lille, September 12-14, 2007.

8. O.T.A. Janssen, S. van Haver, A.J.E.M. Janssen, J.J.M. Braat, H.P. Urbach, S.F. Pereira, "Extended Nijboer-Zernike (ENZ) based mask imaging: efficient coupling of electromagnetic field solvers and the ENZ imaging algorithm," Proc. SPIE 6924, San Jose, February 2008, 692410, pp. 1-9.

9. S. van Haver, O.T.A. Janssen, A.J.E.M. Janssen, J.J.M. Braat, H.P. Urbach, S.F. Pereira, "General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory," Proc. SPIE 6924, San Jose, February 2008, 69240U, pp. 1-8.

10. S. van Haver, J.J.M. Braat, S.F. Pereira, "Enabling aberration retrieval of microlenses with the Extended Nijboer-Zernike (ENZ) diffraction theory," Proc. SPIE 7717, Brussels, May 2010, 77170U, pp. 1-8.

Presentations at scientific meetings

1. P. Dirksen, J.J.M. Braat, P. De Bisschop, A.J.E.M. Janssen, C.A.H. Juffermans, A. Leeuwestein, "Characterization of a projection lens using the extended Nijboer-Zernike approach," SPIE conference on Microlithography, Santa Clara, March 3-8, 2002.

2. J.J.M. Braat, P. Dirksen, A.J.E.M. Janssen, A. van de Nes, "Extended Nijboer-Zernike representation of the field in the focal region of an aberrated high-aperture optical system," poster presented at the Annual Meeting of OSA, Orlando, October 2002.

3. P. Dirksen, J.J.M. Braat, A.J.E.M. Janssen, C. Juffermans, A. Leeuwestein, "Experimental determination of lens aberrations from the intensity point-spread function in the focal region," SPIE conference on Microlithography, Santa Clara, February 23-28, 2003.

4. J.J.M. Braat, P. Dirksen, A.J.E.M. Janssen, A.S. van de Nes, "Extended Nijboer-Zernike description of the high-aperture focal field created by a beam with angular momentum," presented at Focus on Microscopy 2003, Genova, April 13-16, 2003.

5. P. Dirksen, J.J.M. Braat, A.J.E.M. Janssen, A. Leeuwestein, H. Kwinten, D. van Steenwinckel, "Determination of resist parameters using the extended Nijboer-Zernike theory," SPIE conference on Microlithography, Santa Clara, February 21-26, 2004.

6. J.J.M. Braat, P. Dirksen, A.J.E.M. Janssen, A.S. van de Nes, "Quality assessment of focusing optics by aberration retrieval using the extended Nijboer-Zernike diffraction theory," presented at Focus on Microscopy 2004, Philadelphia, April 4-7, 2004.

7. P. Dirksen, J.J.M. Braat, A.J.E.M. Janssen,  D. van Steenwinckel, A. Leeuwestein, "Aberration retrieval for a lithographic lens in the presence of focus variation and spatial diffusion," IISB Lithography Simulation Workshop, Hersbruck, Germany, September 17-19, 2004.

8. J.J.M. Braat, P. Dirksen, A.J.E.M. Janssen, "Extended Nijboer-Zernike analysis for vectorial diffraction calculations and aberration retrieval," IISB Lithography Simulation Workshop, Hersbruck, Germany, September 17-19, 2004.

9. J.J.M. Braat, P. Dirksen, A.J.E.M. Janssen, A.S. van de Nes, "Complex pupil function reconstruction using the extended Nijboer-Zernike diffraction theory,"  abstract presented at OSA Annual Meeting, October 2004, Rochester, USA; corresponding poster presentation: "Complex pupil function reconstruction at high numerical aperture using the extended Nijboer-Zernike diffraction theory".

10. P. Dirksen, J.J.M. Braat, A.J.E.M. Janssen, A. Leeuwestein, "Aberration retrieval for high-NA optical systems using the Extended Nijboer-Zernike theory," SPIE conference on Microlithography, San Jose, February 27 - March 4, 2005.

11. J.J.M. Braat, P. Dirksen, A.J.E.M. Janssen, A. van de Nes, "Polarisation-aberration retrieval for high-NA systems using the extended Nijboer-Zernike diffraction theory," presented at Focus on Microscopy 2005, Jena, March 20-24, 2005.

12. J.J.M. Braat, P. Dirksen, A.J.E.M. Janssen, "Through-focus point-spread function evaluation for lens metrology using the Extended Nijboer-Zernike theory," presented at Fringe 2005, Stuttgart, September 12-14, 2005.

13. J.J.M. Braat, P. Dirksen, A.J.E.M. Janssen, A. Leeuwestein, T. Matsuyama, T. Noda, "Aerial image based lens metrology for wafer steppers," presented at SPIE conference on Microlithography, San Jose, February 19-24, 2006.

14. S. van Haver, J.J.M. Braat, P. Dirksen, A.J.E.M. Janssen, "High-NA lens characterization by through-focus intensity measurements," presented at EOS Annual Meeting 2006 TOM 4, Paris, October 16-19, 2006.

15. S. van Haver, A.J.E.M. Janssen, P. Dirksen, J.J.M. Braat, "Extended Nijboer-Zernike (ENZ) based evaluation of amplitude and phase aberrations on scaled and annular pupils," poster presented at EOS Advanced Imaging Techniques 2007, Lille, September 12-14, 2007.

16. S. van Haver, A.J.E.M. Janssen, P. Dirksen, J.J.M. Braat, "Imaging based on the Extended Nijboer-Zernike (ENZ) formalism," presented at EOS Advanced Imaging Techniques 2007, Lille, September 12-14, 2007.

17. S. van Haver, O.T.A. Janssen, A.M. Nugrowati, J.J.M. Braat, S.F. Pereira, "Novel approach to mask imaging based on the Extended Nijboer-Zernike (ENZ) diffraction theory," poster presented at MNE'07, Copenhagen, September 23-26, 2007. (Received 1st price in best poster award.)

18. O.T.A. Janssen, S. van Haver, A.J.E.M. Janssen, J.J.M. Braat, H.P. Urbach, S.F. Pereira, "Extended Nijboer-Zernike (ENZ) based mask imaging: efficient coupling of electromagnetic field solvers and the ENZ imaging algorithm," presented at SPIE conference on Microlithography, San Jose, February 2008.

19. S. van Haver, O.T.A. Janssen, A.J.E.M. Janssen, J.J.M. Braat, H.P. Urbach, S.F. Pereira, "General imaging of advanced 3D mask objects based on the fully-vectorial extended Nijboer-Zernike (ENZ) theory," presented at SPIE conference on Microlithography, San Jose, February 2008.

20. S. van Haver, O.T.A. Janssen, A.M. Nugrowati, J.J.M. Braat, S.F. Pereira, "Combining various optical simulation tools to enable complex optical system simulations," poster presented at NEMO meeting, Santiago de Compostela, Spain, July, 2008.

21. S. van Haver, O.T.A. Janssen, A.J.E.M. Janssen, J.J.M. Braat, S.F. Pereira, P. Evanschitzky, "Characterization of a novel mask imaging algorithm based on the Extended Nijboer-Zernike (ENZ) formalism," poster presented at MNE conference, Athens, Greece, September, 2008.

22. S. van Haver, O.T.A. Janssen, J.J.M. Braat, S.F. Pereira, "An alternative method for advanced lithographic imaging: the Extended Nijboer-Zernike formalism," lecture at 6th Fraunhofer IISB Lithography Simulation Workshop, Athens, Greece, September, 2008.

23. S. van Haver, O.T.A. Janssen, A.J.E.M. Janssen, J.J.M. Braat, S.F. Pereira, "Image simulations of extended objects using an algorithm based on the Extended Nijboer-Zernike (ENZ) formalism," poster presented at the EOS Annual Meeting 2008, Paris, 29 September - 3 October, 2008. (Received 1st price in best poster award.)

24. S. van Haver, A.J.E.M. Janssen, J.J.M. Braat, S.F. Pereira, "Evaluation of scaled and annular pupils within the frame-work of the Extended Nijboer-Zernike (ENZ) formalism," poster presented at the OSA FiO 2008, Rochester NY, 19 October - 23 October, 2008.

25. S. van Haver, O.T.A. Janssen, J.J.M. Braat, S.F. Pereira, "Characterization of a novel mask imaging algorithm based on the Extended Nijboer-Zernike (ENZ) formalism," poster presented at Micro-Nano Conference, Wageningen, The Netherlands, November, 2008. (Same poster as presented at MNE 2008, see item 21.)

26. S. van Haver, A.J.E.M. Janssen, P. Dirksen, J.J.M. Braat, "Extended Nijboer-Zernike based imaging into an image region containing a layered configuration," presented at EOS Advanced Imaging Techniques 2009, Jena, Germany, June 10-12, 2009.

27. J.J.M. Braat, S. van Haver, S.F. Pereira, "Microlens quality assessment using the Extended Nijboer-Zernike (ENZ) diffraction theory," presented at EOS Optical Microsystems 2009, Capri, Italy, September 27-30, 2009.

28. S. van Haver, J.J.M. Braat, S.F. Pereira, "Accurate and efficient simulation of resist images generated by advanced lithographic systems using the Extended Nijboer-Zernike (ENZ) diffraction theory," presented at Micro-Nano Conference, Delft, The Netherlands, November, 2009.

29. J.J.M. Braat, "Extended Nijboer-Zernike diffraction theory," presented at the SPAM Winterschool, Delft, The Netherlands, January, 2011.

arXiv publications

1. A.J.E.M. Janssen, "Zernike circle polynomials and infinite integrals involving the product of Bessel functions, " arXiv:1007.0667v1 [math-ph] 5 Jul 2010, pp. 1-45.

2. A.J.E.M. Janssen, "A generalization of the Zernike circle polynomials for forward and inverse problems in diffraction theory, " arXiv:1110.2369v1 [math-ph] 11 Oct 2011, pp. 1-43.

3. A.J.E.M. Janssen, "Zernike expansions of derivatives and Laplacians of the Zernike circle polynomials, " arXiv:1404.1766v1 [math-ph] 7 Apr 2014, pp. 1-31.

4. S. van Haver, A.J.E.M. Janssen, "Truncation strategy for the series expressions in the advanced ENZ-theory of diffraction integrals, " arXiv:1407.6589v1 [physics.comp-ph] 24 July 2014, pp. 1-56.

ANZ publications (journal papers and conference proceedings)

1. R.M. Aarts, A.J.E.M. Janssen, "On-axis and far-field sound radiation from resilient flat and dome-shaped radiators," J. Acoust. Soc. Am. 125 (2009), pp. 1444-1455.

2. R.M. Aarts, A.J.E.M. Janssen, "Sound radiation quantities arising from a resilient circular radiator," J. Acoust. Soc. Am. 126 (2009), pp.1776-1787.

3. R.M. Aarts, A.J.E.M. Janssen, "Estimating the velocity profile and acoustical quantities of a harmonically vibrating membrane from on-axis pressure data," presented at NAG/DAGA 2009 Conference, Rotterdam, March 2009.

4. R.M. Aarts, A.J.E.M. Janssen, "Estimating the velocity profile and acoustical quantities of a harmonically vibrating loudspeaker membrane from on-axis pressure data," J. Audio Eng. Soc. 57 (2009), pp.1004-1015.

5. R.M. Aarts, A.J.E.M. Janssen, "Authors' reply to comments on ``Estimating the velocity profile and acoustical quantities of a harmonically vibrating loudspeaker membrane from on-axis pressure data", J. Audio Eng. Soc. 58 (2010), pp.308-310.

6. R.M. Aarts, A.J.E.M. Janssen, "Sound radiation from a resilient spherical cap on a rigid sphere," J. Acoust. Soc. Am. 127 (2010), pp.2262-2273.

7. R.M. Aarts, A.J.E.M. Janssen, "Modeling a loudspeaker as a flexible cap on a rigid sphere," presented at AES 128th Convention, London, May 2010.

8. R.M. Aarts, A.J.E.M. Janssen, "Acoustic holography for piston sound radiation with non-uniform velocity profiles," presented at ICSV 17, Cairo, July 2010.

9. R.M. Aarts, A.J.E.M. Janssen, "Spatial impulse responses from a flexible baffled circular piston," J. Acoust. Soc. Am. 129 (2011), pp.2952-2959.

11. R.M. Aarts, A.J.E.M. Janssen, "Comparing sound radiation from a loudspeaker with that from a flexible spherical cap on a rigid sphere," J. Audio Eng. Soc. 59 (2011), pp.201-212.

12. R.M. Aarts, A.J.E.M. Janssen, "Sound radiation from a loudspeaker, from a spherical pole cap, and from a piston in an infinite baffle," Noise&Vibration Worldwide (April 2012), pp. 12-19.