IRWIN AND JOAN JACOBS CENTER FOR COMMUNICATION AND INFORMATION TECHNOLOGIES Tight Focusing of Wavefronts with Piecewise Constant Phase

The Richards-Wolf approach to analyze tight focusing by high numerical aperture aplanatic optical systems can only be applied to incident waves having a planar (or negligibly curved) wavefront at the entrance pupil. In some cases, however, such as certain singular beams, the incident wave can be represented by a wavefront with, approximately, piecewise constant phase. For wavefronts of this kind we extend the validity of the Richards-Wolf approach to approximate evaluation of the field distribution in the vicinity of the geometrical focus. The proposed method is applied to the derivation of the focal distribution obtained by placing a mask with a π phase shift at the entrance pupil. ©2008 Optical Society of America OCIS codes: (050.1755) Computational electromagnetic methods; (050.1960) Diffraction theory; (260.6042) Singular optics. References and links 1. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems, II Structure of the image field in an optical system”, Proc. R. Soc. Lond. A. 253, 358-379 (1959). 2. A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams”, Optik 41(3), 281-92 (1974) 3. R. Kant, “An analytical solution of vector diffraction for focusing optical systems”, J. of Modern Optics 40(2), 337-347 (1993). 4. P. Torok, 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(2), 325-332 (1995). 5. P. Torok, P. Varga and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field”, J. Opt. Soc. Am. A. 12(10), 2136-2144 (1995). 6. P. Torok, P. Varga and G. Nemeth, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices”, J. Opt. Soc. Am. A. 12(12), 2660-2671 (1995). 7. C. J. R. Sheppard and P. Torok, “Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion”, J. of Modern Optics 44(4), 803-818 (1997). 8. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams”, Opt. Ex. 7(2), 77-87 (2000). 9. C. J. R. Sheppard, “High-aperture beams”, J. Opt. Soc. Am. A. 18(7), 1579-1587 (2001). 10. P. Torok and P.R.T. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy”, Opt. Ex. 12(5), 3605-3617 (2004). 11. S. S. Sherif and P. Torok, “Eigenfunction representation of the integrals of the Debye–Wolf diffraction formula”, J. of Modern Optics 6(15), 857-876 (2005). 12. Gilad M. Lerman and Uriel Levy, “Tight focusing of spatially variant vector optical fields with elliptical symmetry of linear polarization”, Opt. Lett. 32(15), 2194-2196 (2007). 13. B. Spektor, A. Normatov, and J. Shamir, "Singular beam microscopy," Appl. Opt. 47, A78-A87 (to be published in 2008). 14. Y. Li and E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers”, J. Opt. Soc. Am. A 1(8), 801-808 (1984). 15. B. Spektor R. Piestun, and J. Shamir, "Dark Beams with a constant notch", Opt. Letters 21, 456-458 (1996). 16. J. Shamir, Optical Systems and Processes (SPIE, 1999). 17. S. Quabis, R. Dorn, M. Eberler, O. Glockl, G. Leuchs, “The focus of light – theoretical calculation and experimental tomographic reconstruction”, Appl. Phys. B 72, 109–113 (2001).