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Super-Oscillating Airy Pattern Yaniv Eliezer, and Alon Bahabad ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.6b00123 • Publication Date (Web): 17 May 2016 Downloaded from http://pubs.acs.org on May 18, 2016
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ACS Photonics
Super-Oscillating Airy Pattern Yaniv Eliezer∗ and Alon Bahabad Department of Physical Electronics, School of Electrical Engineering, Fleischman Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel E-mail:
[email protected] Abstract We demonstrate both theoretically and experimentally the generation of a tunable two-dimensional superoscillating optical field through the interference of multiple Airy beams. The resulting pattern exhibits self-healing properties for a set of sub-Fourier diffraction spots with decreasing dimensions. Such spatial optical fields might find applications in microscopy, particle manipulation and nonlinear optics.
Keywords Superoscillations, Airy beams, Fourier optics, Self-healing, Diffraction, Interference, Superresolution In 1979 Berry and Balazs 1 have found a unique solution of the Schrodinger equation an infinite non-dispersing wave packet in the form of an Airy function. In 2007 2 Siviloglou et al. have demonstrated its optical analogue in the form of a truncated Airy beam. Airy beams have been widely demonstrated to exhibit unique properties such as weak diffraction, ballistic acceleration 3 and self healing. 4 ∗
To whom correspondence should be addressed
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Various applications have been demonstrated using Airy beams, such as: optical tweezing, 5 generation of curved plasma channels, 6 imaging with extreme field of view 7 and superresolution imaging. 8 Several efforts were made to generate variants of the Airy beam with improved intensity and reduced width of its main lobe. 9,10 The interference of Airy beams was shown in several cases to yield interesting results: Lumer et al. 11 have demonstrated an accelerating Talbot effect while Klein et al. 12 demonstrated the generation of plasmonic hot spots. In 1988 Aharonov et al., 13 while developing the concept of quantum weak measurements, have found a family of functions exhibiting rapid local oscillations exceeding the functions’ highest Fourier component. Such functions are commonly known today as superoscillating functions. In 2006, Berry and Popescu 14 have introduced superoscillations into optics, suggesting that super-oscillating optical beams can obtain super-resolution without evanescent waves. This suggestion was later realized experimentally in a series of works demonstrating optical super-resolution. 15–17 The same concept was also adopted in the time domain to suggest overcoming absorption in dielectric materials, 18 for realizing sub-Fourier focusing of radio-frequency signals 19 and for achieving temporal optical super-resolution. 20 A nondiffracting super-oscillating optical beam was also demonstrated. 21 In this work, we theoretically and experimentally demonstrate that a judicious selection of interfering Airy beams (modes) can create a Superoscillatory Airy Pattern (SOAP) exhibiting unique properties. In particular, the SOAP contains oscillations which are faster than those of its highest Airy mode leading to an ordered array of sub-Fourier focused light spots. In addition, the pattern is resilient (to some extent) to obstruction through the self-healing property of its constituting modes.
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Theory It was shown by Berry and Popescu that the complex function: 14
fSO (t) = [cos (K0 x) + ia sin (K0 x)]N ,
a > 1,
N ∈ N+
(1)
superoscillates: while the highest Fourier component of this function is N K0 , around x ≈ 0 it oscillates a times faster:
fSO (x π/K0 ) ≈ exp [N log (1 + iaK0 x)] ≈ exp (iaN K0 x)
(2)
It is possible to express the imaginary part of Eq. 1 as a sum of discrete sine modes having a specific set of amplitudes and frequencies Cqn and qn K0 correspondingly: 18 N X
N Im{fSO (x)} = N −2 a 2 k k=−N N −k X k X k i[2(l+m)−N ]K0 x m N −k · (−1) e l m l=0 m=0 i
k
N
=
b2c X
Cqn sin (qn K0 x)
(3)
n=0
where k ∈ {odd}, K0 is an arbitrary fundamental spatial frequency, qn ≡ 2n + µN and µN ≡ mod(N, 2) (where mod stands for the modulo operation). In the present case instead of using sine functions as the modes for constructing a superoscillating function we use a basis of finite Airy functions:
f (x) =
X An n
αn
Ai
x αn
exp (γx)
(4)
where Ai(x) is the Airy function in the variable x, γ is a common decay rate, An are relative amplitude coefficients and αn determine the rate of oscillation of each Airy beam. Using
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Stokes’ integral approximation for the Airy function 22 it is possible to approximate Eq. 4 to a sum of chirped sine modes decaying with a common envelope: X 1 3 1 An αn − 4 sin f (x) ∼ = √ x− 4 exp (γx) π n
2 −3 3 π αn 2 x 2 + 3 4
(5)
Now we can set An and αn according to Eq.3: αn =
3 Kn 2
− 32 ;
3 4
An = Cqn αn = Cqn
3 Kn 2
− 12 (6)
where Kn = qn K0 . These assignments result in a function f (x) which is chirped and superoscillatory as long as
π 4
