Natural superoscillations in monochromatic waves in D dimensions

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1 IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. (9) 3 (8pp) doi:.88/75-83///3 FAST TRACK COMMUNICATION Natural superoscillations in monochromatic waves in D dimensions M V Berry and M R Dennis H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 TL, UK Received 3 October 8, in final form November 8 Published December 8 Online at stacks.iop.org/jphysa//3 Abstract For monochromatic waves satisfying the Helmholtz equation with wavenumber k, superoscillations correspond to local wavenumbers (magnitude of phase gradient) greater than k. Large values of local wavenumber are associated with phase singularities. For isotropic random waves (superpositions of many nonevanescent waves) in D dimensions, we show that the probability that a point in the field is superoscillatory increases from.93 to.39 as D increases from to infinity. The peculiar case D = is eamined in detail. PACS numbers:.3.nw, 3.65.Vf,.5.Hz,.3.Kq,.3.Ms. Introduction It is now understood that band-limited functions can oscillate faster than their fastest Fourier components [, ]. This apparently paradoical behaviour is demystified by the observation, now well understood [3], that such superoscillations occur in regions where the amplitude of the function is small. Several constructions have been devised, leading to functions that superoscillate arbitrarily rapidly in arbitrarily large domains. Here we will consider superoscillations occurring naturally in band-limited functions of a special but physically important type: monochromatic comple scalar waves in D dimensions. These have the form ψ(r) = u(r) +iv(r) = ρ(r) ep{iχ(r)}, r ={,..., D }, (.) with ψ satisfying the Helmholtz equation ψ + k ψ =, (.) in which k is the free-space wavenumber. Superoscillations will correspond to ψ varying on subwavelength scales, that is, scales smaller than the wavelength π/k. A natural measure of the oscillation at r is the rate at which the phase χ(r) is changing; this is the local wavenumber: k(r) χ(r) =Im[ log ψ(r)] = u(r) v(r) v(r) u(r) ρ(r). (.3) 75-83/9/3+8$3. 9 IOP Publishing Ltd Printed in the UK

2 Figure. Superoscillations for D =, calculated from the plane-wave superposition (.5)for N = and k = π (i.e. wavelength λ = ), showing wavefronts at phase intervals π/, intersecting at the vortices (black lines) and the superocillatory region where k(r) > k (white). There are two situations in which k(r) can eceed k. In the first, the decomposition of ψ into plane waves includes evanescent waves. A two-dimensional eample is ψ(r) = ep{ik} ep { y K k}, K > k. (.) Obviously, k(r) = K: there are subwavelength oscillations transverse to the decaying direction; these are not superoscillations, because the fast variation is present in the spectrum of ψ. We are interested in the contrary case, where ψ contains only real plane waves; then regions where k(r) > k correspond to genuine superoscillations. We epect these regions to be centred on wave vortices [ 7], that is, the phase singularities, where the functions vanish and the phase varies infinitely rapidly. By constructing waves with a cluster of many close-lying vortices, the superoscillations can be made faster and more numerous [7]. We seek to quantify the degree of superoscillation not in artificial constructions but in naturally occurring monochromatic waves. We model these by isotropic superpositions of many plane waves: N ψ(r) = a n ep{ik n r}, N, k n =k, (.5) n= in which the amplitudes a n are random comple numbers and the wavevector directions k n /k are uniformly distributed. Figures and depict sample functions for D = and D = 3. As epected, the regions of superoscillation (highlighted) include the wave vortices. The simplest measure of superoscillation is the probability P D super that k(r) > k for a randomly chosen point r. We will calculate this for general D in section, using the fact

3 Figure. Superoscillations for D = 3, for a superposition (.5) with N =. The shaded surface is the boundary k(r) = k of the superoscillatory region, which includes the phase singularities (black lines). The volume displayed is one cubic wavelength. (a consequence of the central limit theorem) that the waves defined by (.5) are isotropic Gaussian random functions. This generalizes the recent result P super = /3 by one of us [8]. The case D = is peculiar, and deserves a special discussion (section 3).. Superoscillation probability The desired quantity is P D super = dkp D (k), (.) k where P D (k) is the probability distribution of k(r)forwavesinddimensions. From (.3), this is ( ) P D (k) = D k δ D u v v u k ρ = Dk D { } (.) d D u v v u s ep { ik s} ep is, (π) D ρ where π D/ D = Ɣ ( (.3) D) is the surface area of the unit sphere in D dimensions, and denotes averaging over u, v and u, v. We use the following facts: u and v are statistically independent of u and v; u and v are independent of each other and have the same distribution; u and v are independent 3

4 . P D k/k Figure 3. Probability distributions P D (k) (equation (.7)) for the indicated values of D. of each other and have the same distribution; and for isotropic randomness, the components i u and j u are independent of each other for i j and have the same distribution. Thus, evaluating the averages over the gradients first, and in an obvious notation, ep { is u v v u ρ } = = ep { ρ (s (cos χ v sin χ u)) u, v ρ,χ }. (.) { ep ρ s ( u) From isotropy and monochromaticity follows ( u) = D u = k D u, (.5) and so, performing the average over u and v by integrating over ρ and χ, P D (k) = Dk D D D/ (π) D/ k D ρ D ep { Dk ρ } u D/ k u Thus D k D D D/ = (π) D/ k D u D/+ P D (k) = ρ,χ ρ,χ ( dρρ D+ ep { ρ Dk )} u k +. (.6) k kd ( ) k + k/. (.7) D D+ Several of these distributions are shown in figure 3. In the limit D, { } P (k) = k k ep k. (.8) 3 k D Now (.) gives our main result: the superoscillation probability ( ) D D/ P D super =. (.9) D + }

5 These numbers slowly increase with D; a few values are P super = =.9 89, P super = 3, P 3 super = =.35 8,..., P super = e = (.) The distributions (.7) decay as k 3 for large k, so moments of order or higher diverge. The first moment the mean value of k(r) is π Ɣ ( k D = dkkp D (k) = (+D)) D Ɣ (, (.) D) and increases from to π/ asd increases from to infinity. 3. Superoscillations for D = For this section only, it is convenient to define the local wavenumber k() without the modulus sign in (.3), so that it can take positive as well as negative values, that is k() = χ() = Im[ log ψ ()]. (3.) The formulae in the previous section have sensible limits when D =. This might seem strange, because the plane-wave components of a monochromatic wave ψ() are concentrated at k n = +k and k n = k, and it is hard to see how such a function can superoscillate. It would seem that the local wavenumber k() should be +k if the positive amplitude dominates, or k if the negative amplitude dominates. Nevertheless, monochromatic waves in one dimension can superoscillate, as we show now. It is necessary to take the limit of a nonmonochromatic wave as its spectrum is concentrated onto wavenumbers +k and k, that is N ψ() = a n ep{ik n }, n= k k n k ( δ) for n N, (3.) k ( δ) k n k for N + n N, N, δ. As numerical simulation indicates (figure ), the distribution of local wavenumbers matches the theoretical distribution very well. And the superoscillation fractions for superpositions of N = 5 waves, with 5 wavenumbers randomly distributed between k and.99k and 5 randomly distributed between.99k and k (i.e. δ =. in (3.)), typically agree with the theoretical value P super =.9 89 to better than % in runs of 5 samples. A sample wave of this type, together with the corresponding local wavenumber k(), is shown in figure 5. It is clear that the large values of k() occur close to the places where ψ() is small places that in more dimensions would be the phase singularities. To understand how this generates the asymptotic superoscillatory distribution P (k) k /k3 for k k (cf (.7)), it will suffice to study the simple two-wave model, ψ() = (+ε) ep{i} +ep{ i}, (3.3) where ε is a real number (any change in the phase of the amplitude can be accommodated by shifting ). From (3.), the local wavenumber is ε + ε k() = ε + cos. (3.) 5

6 . P k/k Figure. Dashed curve: probability distribution P (k), from simulation of 5 samples of the wave (3.) with N = 5, δ =.; thin curve (barely visible behind): theoretical distribution (.7). ψ 6 (a) k 3 (c) ψ (b) k 8 6 (d) Figure 5. Sample wave (3.) ford =, with N = 5, δ =., k =. (a) Thin curve, Re ψ(); dashed curve, Im ψ(); thick curve, ψ(). (b) Magnification of (a). (c) Local wavenumber k() corresponding to (a). (d) Magnification of (c). As figure 6 illustrates (for a negative value of ε), the maima of k(), of magnitude /ε, coincide with the minima of ψ(), of magnitude ε,at = π/(modπ). We are interested in large values of k(), for which 6 ( ) k + ξ ε 8ξ ε. (3.5) 3

7 ψ (a) - - k (b) Figure 6. (a) Two-wave superposition (3.3) with ε =.: thin curve, Re ψ(); dashed curve, Im ψ(); thick curve, ψ(). (b) Local wavenumber k() (3.) corresponding to (a). For fied ε, this basic narrow-parabola model gives the probability distribution of the local wavenumber as P(ε,k) = ε/ ( ( d ξδ k ε ε 8ξ )) ( ε = k ε 3 ). (3.6) kε ε In the Gaussian random waves ψ we are interested in, the minimum values ε of ψ are not fied but fluctuate. Therefore, we can estimate P (k) by averaging over ε, using the fact that the distribution of minimum values vanishes linearly as ε. Thus P (k) dεεp(ε,k) = 5k (8 ( k) k(8+k +3k )) 3 = 6 (3.7) if k>, 5k 3 thereby eplaining the superoscillations for D =.. Discussion We have shown that typical waves superoscillate over surprisingly large regions of D- dimensional space, increasing from a fraction.93 for D = to.39 as D. This 7

8 reinforces and etends the result /3 recently obtained [8] ford =. In comparison with the etreme superoscillations that can be artificially constructed [], those considered here are relatively modest: typically a single large spike in k(r) asr passes a nearby phase singularity. Nevertheless, the probability distribution of k(r) has a long superoscillatory tail, slowly decaying as k 3. Beyond statistics, much remains to be eplored about the geometry and topology of superoscillatory regions. Preliminary evidence for D = suggests that these regions are usually connected, including infinitely many wave vortices in percolating clusters. The same could be true for D >, but we do not know. We have concentrated on the superoscillations, for which k(r) > k, and the related large-k asymptotics of P D (k). However, we note that for k, that is suboscillations (phase variations much slower than the common wavelength λ), P D (k) vanishes as k D (equation (.7)), and is nonanalytic for D = (equation (.8)). Wave superoscillations have quantum implications that will be eplored elsewhere. In brief, large values of k(r) could be measured in a quantum weak measurement [9], as highmomentum impulses from quanta (photons in the case of optical fields) detected by detectors small compared with the wavelength, for eample narrow slits []. Momentum arises because k(r) can be regarded as the local epectation value of the momentum operator. In an obvious quantum notation, k(r) = ψ ˆp(r) ψ, (.) where ˆp(r) = (δ(ˆr r) ˆp +ˆpδ(ˆr r)), (.) and, in position representation, ˆp = i. (.3) Acknowledgments We thank the organizers of the meeting SO8 (Alushta, Crimea), where this work was completed. MVB is supported by the Leverhulme Trust, and MRD by the Royal Society. MRD thanks Dr Johannes Courtial for discussions and insight at the early stages of this work. References [] Berry M V 99 Faster than Fourier in quantum coherence and reality Celebration of the 6th Birthday of Yakir Aharonov ed J S Anandan and J L Safko (Singapore: World Scientific) pp [] Berry M V and Popescu S 6 Evolution of quantum superoscillations, and optical superresolution without evanescent waves J. Phys. A: Math. Gen [3] Ferreira P J S G and Kempf A 6 Superoscillations: faster than the Nyquist rate IEEE Trans. Signal Process [] Nye J F and Berry M V 97 Dislocations in wave trains Proc. R. Soc. A [5] Nye J F 999 Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Bristol: Institute of Physics Publishing) [6] Berry M V and Dennis M R Phase singularities in isotropic random waves Proc. R. Soc. A [7] Berry M V 8 Waves Near Zeros in Coherence and Quantum Optics ed N P Bigelow, J H Eberly and C R J Stroud (Washington, DC: Optical Society of America) pp 37 [8] Dennis M R, Hamilton A C and Courtial J 8 Superoscillation in speckle patterns Opt. Lett. arxiv:8.98 [physics.optics] at press [9] Aharonov Y and Rohrlich D 5 Quantum Paradoes: Quantum Theory for the Perpleed (Weinheim: Wiley- VCH) [] Kempf A and Ferreira P J S G Unusual properties of superoscillating particles J. Phys. A: Math. Gen