Minimizing the bright/shadow focal spot size for differently polarized incident waves through the additional apodization of the focusing system output pupil by use of an optical element with the vortex phase dependence on angle and the polynomial amplitude dependence on radius is studied. The coefficients of the radial polynomial were optimized with the aim of fulfilling certain conditions such as the energy efficiency preservation and keeping the side lobes under control. The coefficients were chosen so as to minimize the functional using Brent’s method. 1. Introduction Recent years have seen the publication of a large number of articles dealing with obtaining a smaller transverse size of the focal spot using a high focusing system [1–5]. The smallest transverse size of the focal spot has been obtained for a radially polarized beam in which the longitudinal component makes the maximal contribution to the total intensity. There is a variety of techniques to gain the contribution of the longitudinal component to the total intensity, including the use of an annular aperture which allows only the peripheral part of radiation to pass through the lens [1, 2] and the use of additional phase optical elements, which provides a higher energy efficiency [3–5]. However, this study is concerned with not simply controlling the contribution of different electric field components into the focal spot region [5], but rather minimizing the (bright/dark) focal spot size by optimally selecting the pupil’s transmission function. In [6], it was proposed that the focal spot formed by a high focusing system should be reduced in size using radially polarized Laguerre-Gauss modes of higher radial order, devoid of vortex phase component. The positive role of the vortex phase function which allows obtaining a smaller focal spot in individual components of the sharply focused electric field at different polarization types was analyzed in [4], whereas the possibility of obtaining a smaller focal spot in terms of total intensity by introducing additional variations of radius was shown in [5]. The use of the transmission functions in the form of Zernike polynomials, including those containing vortex phase dependence, was considered in [7]. Thus, it has become possible to simultaneously introduce vortex phase dependence and amplitude variations on radius. Note that it has also been shown that it is possible to reduce not only the bright spot but the shadow region as well. In areas such as optical trapping and micromanipulation [8], STED-microscopy [9], and shadow microscopy [10], there
References
[1]
S. Quabis, R. Dorn, M. Eberler, O. Gl？ckl, and G. Leuchs, “Focusing light to a tighter spot,” Optics Communications, vol. 179, no. 1, pp. 1–7, 2000.
[2]
R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Physical Review Letters, vol. 91, no. 23, pp. 2339011–2339014, 2003.
[3]
L. E. Helseth, “Smallest focal hole,” Optics Communications, vol. 257, no. 1, pp. 1–8, 2006.
[4]
S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system,” Journal of Modern Optics, vol. 58, no. 9, pp. 748–760, 2011.
[5]
S. N. Khonina and S. G. Volotovsky, “Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures,” Journal of the Optical Society of America A, vol. 27, no. 10, pp. 2188–2197, 2010.
[6]
Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” Journal of the Optical Society of America A, vol. 24, no. 6, pp. 1793–1798, 2007.
[7]
S. N. Khonina, A. V. Ustinov, and E. A. Pelevina, “Analysis of wave aberration influence on reducing focal spot size in a high-aperture focusing system,” Journal of Optics, vol. 13, no. 9, Article ID 095702, 2011.
[8]
J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Optics Letters, vol. 25, no. 4, pp. 191–193, 2000.
[9]
S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Optics Letters, vol. 19, no. 11, pp. 780–782, 1994.
[10]
D. P. Biss, K. S. Youngworth, and T. G. Brown, “Dark-field imaging with cylindrical-vector beams,” Applied Optics, vol. 45, no. 3, pp. 470–479, 2006.
[11]
S. N. Khonina, N. L. Kazanskiy, and S. G. Volotovsky, “Influence of vortex transmission phase function on intensity distribution in the focal area of high-aperture focusing system,” Optical Memory and Neural Networks, vol. 20, no. 1, pp. 23–42, 2011.
[12]
R. P. Brent, Algorithms for Minimization without Derivatives, Prentice-Hall, New York, NY, USA, 1973.
[13]
M. Abramovitz and I. A. Stegun, Eds., Handbook of Mathematical Functions, Applied Math. Series, National Bureau of Standards, 1965.
[14]
S. N. Khonina and I. Golub, “Enlightening darkness to diffraction limit and beyond: comparison and optimization of different po-larizations for dark spot generation,” Journal of the Optical Society of America A, vol. 29, no. 7, pp. 1470–1474, 2012.
[15]
Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Advances in Optics and Photonics, vol. 1, pp. 1–57, 2009.
[16]
S. N. Khonina and I. Golub, “Optimization of focusing of linearly polarized light,” Optics Letters, vol. 36, no. 3, pp. 352–354, 2011.
[17]
B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proceedings of the Royal Society A, vol. 253, pp. 358–379, 1959.
[18]
http://alglib.sources.ru/.
[19]
S. N. Khonina and A. V. Ustinov, “Sharper focal spot for a radially polarized beam using ring aperture with phase jump,” Journal of Engineering, vol. 2013, Article ID 512971, 8 pages, 2013.
[20]
T. R. M. Sales and G. M. Morris, “Diffractive superresolution elements,” Journal of the Optical Society of America A, vol. 14, no. 7, pp. 1637–1646, 1997.
[21]
J. Bewersdorf, A. Egner, and S. W. Hell, “4pi-confocal microscopy is coming of age,” G.I.T. Imaging & Microscopy, vol. 4, pp. 24–25, 2004.
[22]
L. E. Helseth, “Breaking the diffraction limit in nonlinear materials,” Optics Communications, vol. 256, no. 4–6, pp. 435–438, 2005.
[23]
G. T. di Francia, “Degrees of freedom of an image,” Journal of the Optical Society of America, vol. 59, no. 7, pp. 799–804, 1969.
[24]
F. M. Huang and N. I. Zheludev, “Super-resolution without evanescent waves,” Nano Letters, vol. 9, no. 3, pp. 1249–1254, 2009.
[25]
T. Grosjean and D. Courjon, “Photopolymers as vectorial sensors of the electric field,” Optics Express, vol. 14, no. 6, pp. 2203–2210, 2006.
[26]
A. P. Prudnikov, A. Yu. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 2. Special Functions, Gordon & Breach Sci. Publ., New York, NY, USA, 1990.
