Journal of the European Optical Society - Rapid publications, Vol 4 (2009)

Image formation in a multilayer using the Extended Nijboer-Zernike theory

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, S. F. Pereira

Abstract


We study the image formation by a high-numerical-aperture optical imaging system in the presence of a multilayer structure in the region around the image plane. Earlier references to this subject in the literature use numerical solutions of the diffraction integrals. In this paper, we use a numerical approach based on the semi-analytic Extended Nijboer-Zernike (ENZ) theory to solve the diffraction integrals in the presence of a multilayer structure. The specific ENZ calculation scheme uses the complex Zernike expansion of the complex amplitudes of forward and backward propagating plane wave components in a certain layer of the multilayer stack. By its nature, the ENZ approach enables an accurate and fast calculation of the vector field in the stratified image region. Examples of multilayer imaging that are encountered in high-numerical-aperture optical systems and in optical lithography for semiconductor manufacturing are presented and the accuracy of the ENZ approach is examined.

© The Authors. All rights reserved. [DOI: 10.2971/jeos.2009.09048]

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References


V. S. Ignatowsky, "Diffraction of a lens of arbitrary aperture" Trans. Opt. Inst. 1, 1-36 (1919).

E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image" Proc. Roy. Soc. London A 253, 349-357 (1959).

B. Richards, and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system" Proc. Roy. Soc. London A 253, 358-379 (1959).

H. Ling, and S. Lee, "Focusing of electromagnetic waves through a dielectric interface" J. Opt. Soc. Am. A 1, 965-973 (1984).

P. Trk, P. Varga, Z. Laczik, and G. R. Booker, "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation" J. Opt. Soc. Am. A 12, 325-332 (1995).

A. Egner, and S. W. Hell, "Equivalence of the Huygens-Fresnel and Debye approach for the calculation of high aperture point-spread functions in the presence of refractive index mismatch" J. Microsc. 193, 244-249 (1999).

C. Sheppard, and C. Cogswell, "Effects of aberrating layers and tube length on confocal imaging properties" Optik 87, 34-38 (1991).

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, "Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index" J. Microsc. 169, 391-405 (1993).

D. G. Flagello, and T. D. Milster, "Three-dimensional modeling of high-numerical-aperture imaging in thin films" Proc. SPIE 1625, 246-261 (1992).

D. G. Flagello, T. Milster, and A. E. Rosenbluth, "Theory of high-NA imaging in homogeneous thin films" J. Opt. Soc. Am. A 13, 53-64 (1996).

A. K. Wong, Optical Imaging in Projection Microlithography (SPIE Press, 2005).

M. Mansuripur, "Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 [1] 2 matrices" J. Appl. Phys. 67, 6466-6475 (1990).

M. Mansuripur, "Effects of high-numerical-aperture focusing on the state of polarization in optical and magneto-optical data storage systems" Appl. Opt. 30, 3154-3162 (1991).

A. S. van de Nes, L. Billy, S. F. Pereira,and J. J. M. Braat, "Calculation of the vectorial field distribution in a stratified focal region of a high numerical aperture imaging system" Opt. Exp. 12, 1281-1293 (2004).

J. M. A. van den Eerenbeemd, D. M. Bruls, C. A. Verschuren, B. Yin, and F. Zijp, "Towards a Multi-Layer Near-Field Recording System: Dual-Layer Recording Results" Jpn. J. Appl. Phys. 46, 3894-3897 (2007).

C. J. R. Sheppard, T. J. Connolly, J. Lee, and C. J. Cogswell, "Confocal imaging of a stratified medium" Appl. Opt. 33, 631-640 (1994).

A. J. E. M. Janssen, "Extended Nijboer-Zernike approach for the computation of optical point-spread functions" J. Opt. Soc. Am. A 19, 849-857 (2002).

J. Braat, P. Dirksen, and A. J. E. M. Janssen, "Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions" J. Opt. Soc. Am. A 19, 858-870 (2002).

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

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

J. Lee, M. van der Aa, C. Verschuren, F. Zijp, and M. van der Mark, "Development of an air gap servo system for high data transfer rate near-field optical recording" Jpn. J. Appl. Phys. 44, 3423-3426 (2005).

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

A. J. E. M. Janssen, and P. Dirksen, "Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform" J. Europ. Opt. Soc. Rap. Public. 2, 07012 (2007).

A. J. E. M. Janssen, and P. Dirksen, "Concise formula for the Zernike coefficients of scaled pupils" J. Microlithogr. Microfabr. Microsyst. 5, 030501 (2006).

M. Abramowitz, and I. A. Stegun, Handbook of Mathematical Functions, 9th Edition. (Dover Publications Inc., New York, 1972).

G. E. Andrews, R. Askey, and R. Roy, "Special Functions" in Encyclopedia of Mathematics and its Applications, Vol. 71 (Cambridge University Press, Cambridge, 1999).