Exact nonparaxial transmission of subwavelength detail using superoscillations

Any object, described by a target function that need not be band-limited, can be sampled at any chosen set of points and then propagated without evanescent waves, using a function which is band-limited, so as to be imaged exactly (i.e. nonparaxially) at multiples of a given repetition distance. If the samples span a sub-wavelength region, the repeated images are superoscillatory. The number N of samples is equal to the repetition distance measured in wavelengths and also to the number of plane waves in the propagating field. If N ? 1, the waves form a quasi-continuum, and asymptotics enables an almost-explicit description of the superoscillations. But the matrix involved is ill-conditioned (many of its eigenvalues are very small), so this method of sub-wavelength imaging would be pathologically sensitive to noise, and the depth of focus is exponentially small.

[1]  Yonina C. Eldar,et al.  Super-resolution and reconstruction of sparse sub-wavelength images: erratum , 2010 .

[2]  M. V. Berry,et al.  Integer, fractional and fractal Talbot effects , 1996 .

[3]  W. Duane Montgomery,et al.  Self-Imaging Objects of Infinite Aperture* , 1967 .

[4]  S G Lipson,et al.  Superresolution in far-field imaging. , 2002, Journal of the Optical Society of America. A, Optics, image science, and vision.

[5]  Ronald F. Boisvert,et al.  NIST Handbook of Mathematical Functions , 2010 .

[6]  D. Struppa,et al.  On the Cauchy problem for the Schrödinger equation with superoscillatory initial data , 2013 .

[7]  A. Kempf,et al.  Analysis of superoscillatory wave functions , 2004 .

[8]  G. Toraldo di Francia,et al.  Super-gain antennas and optical resolving power , 1952 .

[9]  Henry Cox,et al.  Practical supergain , 1986, IEEE Trans. Acoust. Speech Signal Process..

[10]  P. Hemmer,et al.  The universal scaling laws that determine the achievable resolution in different schemes for super-resolution imaging , 2012 .

[11]  M. Berry.  Superoscillations, Endfire and Supergain , 2014 .

[12]  P.J.S.G. Ferreira,et al.  Superoscillations: Faster Than the Nyquist Rate , 2006, IEEE Transactions on Signal Processing.

[13]  Yonina C Eldar,et al.  Super-resolution and reconstruction of sparse sub-wavelength images. , 2009, Optics express.

[14]  Yonina C. Eldar,et al.  Super-resolution and reconstruction of sparse images carried by incoherent light. , 2010, Optics letters.

[15]  J. Anandan,et al.  Quantum Coherenece and Reality, In Celebration of the 60th Birthday of Yakir Aharonov , 1995 .

[16]  Sandu Popescu,et al.  Evolution of quantum superoscillations, and optical superresolution without evanescent waves , 2006 .

[17]  Vaidman,et al.  How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. , 1988, Physical review letters.

[18]  G. Hardy,et al.  Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work , 1978 .

[19]  J. Bucklew,et al.  Theorem for high-resolution high-contrast image synthesis , 1985 .

[20]  A. M. H. Wong,et al.  Superoscillatory Radar Imaging: Improving Radar Range Resolution Beyond Fundamental Bandwidth Limitations , 2012, IEEE Microwave and Wireless Components Letters.

[21]  George V Eleftheriades,et al.  Adaptation of Schelkunoff's Superdirective Antenna Theory for the Realization of Superoscillatory Antenna Arrays , 2010, IEEE Antennas and Wireless Propagation Letters.

[22]  Daniele C. Struppa,et al.  Some mathematical properties of superoscillations , 2011 .

[23]  Mark R. Dennis,et al.  Aberration-like cusped focusing in the post-paraxial Talbot effect , 2012, 1204.5994.

[24]  A. Lohmann,et al.  Optical wave fields with lateral and longitudinal periodicity. , 2009, Applied optics.

[25]  Yakir Aharonov,et al.  Quantum Paradoxes: Quantum Theory for the Perplexed , 2005 .

[26]  S. Schelkunoff.  A mathematical theory of linear arrays , 1943 .

[27]  Mark R. Dennis,et al.  A super-oscillatory lens optical microscope for subwavelength imaging. , 2012, Nature materials.

[28]  Jürgen Jahns,et al.  Fractional Montgomery effect: a self-imaging phenomenon. , 2005, Journal of the Optical Society of America. A, Optics, image science, and vision.

[29]  D. Psaltis,et al.  Superoscillatory diffraction-free beams , 2011 .

[30]  R. P. Haviland,et al.  Supergain antennas: possibilities and problems , 1995 .

[31]  Achim Kempf,et al.  Unusual properties of superoscillating particles , 2004 .

[32]  Jani Tervo,et al.  Exact self-imaging of transversely periodic fields. , 2004, Journal of the Optical Society of America. A, Optics, image science, and vision.

[33]  J. Bucklew,et al.  Synthesis of binary images from band-limited functions , 1989 .

[34]  Jari Lindberg,et al.  Mathematical concepts of optical superresolution , 2012 .

[35]  Jiri Kyvalsky,et al.  The self-imaging phenomenon and its applications , 2003, Photonics Prague.

[36]  A. Baskakov,et al.  Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations , 2013 .