Three universal oscillatory asymptotic phenomena

1. Three universal oscillatory asymptotic phenomena Michael Berry Physics Department University of Bristol United Kingdom http://www.phy.bris.ac.uk/staff/berry_mv.html

2. function f(z) with two comparably contributing exponentials for large values of some parameter (not indicated) Stokes line , f+ dominates antistokes line 1. Dominance by subdominant exponentials (DSE) z plane exponentials comparable

3. Fplane stokes line subdominant dominant antistokes line

4. subdominant dominant very common situation: a+ and a-depend differently on z(and hence on F); ignoring logarithms, no loss of generality with m>0 dominance by subdominant exponential (DSE) if

5. universal curve stokes line DSE zeros lie on DSE boundary antistokes line DDE

6. dominant subdominant (for Imz>0) example 1: saddle-point and end-point end-point at t=z, saddle at t=0

7. phase contours of g1(F)

8. ReF ReF subdominant DSE DSE ImF ImF dominant

9. example 4: suppressed end-point saddle at t=z, end-point at z=0, suppressed but dominant for Imz>0

10. application of f4(z) to conical diffraction emerging from the slab, an axial spike and a cylinder of light, radius R0 Hamilton,Lloyd,Poggenforff, Raman… focal image plane crystal slab entrance face light incident on a slab of crystal with three different principal dielectric constants wavelength l M V Berry 2004 ‘Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike’, J. Opt. A 6, 289-300

11. focal image geometrical optics diffraction where increasing Z x (ro=50) far field conical-diffraction ring profile is

12. secondary rings as interference between geometrical optics (saddle), and wave scattered from conical point (end-point) again, main contribution from subdominant exponential (saddle)

13. seek z plane t C 2. Universal attractor of differentiation as nincreases, “all smooth functions look like cost “ (cf. orthogonal polynomials)

14. singularities (poles or branch points) closest to real axis Imz t1++it2+ Rez t t1--it2- C if z(t) analytic in a strip including the real axis:

15. assume for simplicity that t2+≠t2- (e.g. z(t) not real), and define t2 =min(t2±) then, up to a constant and (real) shift of torigin, m depends on the type of singularity as t increases from - to +, zn(t) describes Maclaurin’s sinusoidal spiral (1718) (universal loops); in polar coordinates,

16. the nth loop has n/2+1 windings Imz Rez z2 z1 z0 z500 z9 z50 universal attractor of hodograph map (‘velocity’ dz/dt)

17. Rez0(t) Rez2(t) Rez10(t) Rez100(t) Rez500(t) Rez50(t) as n increases, “universal cosine” emerges near t=0: instability of differentiation?

18. universal loops in geometric phase physics: state vector Y driven by slowly-varying operator H(et) (adiabatic quantum mechanics) geometric phase, depending on geometry of H loop geometric phase corrections as t increases from - to , H describes a loop in operator space Y starts in an eigenstate of H(± ) for small e, Y returns to its original form, apart from a phase factor:

19. geometric phase corrections are determined by time-dependent transformation to eigenbasis of H(et), giving H1(t), and then iterating to H2(t), etc; phase corrections are related to areas of Hnloops in the simplest nontrivial case, iterated loops are given by the derivative (hodograph) map: phase corrections get smaller and then inevitably diverge, universally

20. if z(t) has no finite singularities, asymptotics of zn(t) determined by saddle-points in integral representation, e.g. example (suggested by David Farmer): inverse gamma derivatives 1/G(z) has zeros at z=0, -1, -2…

21. as n increases, zeros migrate into the positive z axis:

22. has, conventionally, a fastest variation 3. Superoscillations a band-limited function nevertheless - counterintuitively - such functions can oscillate arbitrarily faster thankmax, over arbitrarily large intervals: they can ‘superoscillate’ several recipes, suggested by Aharonov, Popescu…

23. suggests a -function, centred on thecomplex positioniA: so consider the band-limited function kmax u -function, centred on u=A: where k(u) is even with k(0)=kmax and |k(u)|≤kmax for real u (Aharonov suggested k(u)=kmaxcosu)

24. the complex -function argument suggests that, for small , which superoscillates where saddle at us(,A), where then more careful small- analysis: saddle-point method, with =x2

25. Imus increasing Reus small , us=iA large x, us0 superoscillations normal oscillations, exponentially larger deportment of saddle us(,3)

26. kloc() cosh2 A=2  local wavenumber

27. log10Ref x x x superoscillations, period sech4=0.037 normal oscillations, period 1 A=4, =0.2

28. Ref exact saddle-point A=2 (A≤2) x log|Ref| Ref x x test of saddle-point approximation with k(u)=1-u2/2, |u|≤2

29. x/ 0 x/ 0 demystification: a simple function with rudimentary superoscillations cosx+1 band-limited, with frequencies 0,+1,-1 cosx+1- pairs of close zeros, separated by 

30. period x= wavenumber aN if x<<1, but in the fourier series |km|≤1 a periodic superoscillatory function (with Sandu Popescu) superoscillations with a factor a faster than normal oscillations

31. involving local wavenumber a=4 k superoscillations normal oscillations suboscillations x alternative form:

32. number of superoscillations: fast superoscillations near x=0 a=4 normal oscillation wavelength number of fast superoscillations

33. superoscillations invisible in power spectrum: narrow spectrum, centred on k=1/a

34. generalization: integrate over a to generate arbitrary functions locally, as band-limited function example: a locally narrow gaussian narrower than cos(Nx) if A>1

35. g(x) cos(40x) A=4, N=40 locally, g~exp{-160x)2/2}

36. in signal processing, superoscillations in a function f(t) emerging from a perfect low-pass filter could generate the illusion that the filter is leaky Beethoven’s 9th symphony - a one-hour signal, requiring up to 20kHz for accurate reproduction - can be generated by a 1Hz signal (but after the hour, f(t) rises by a factor exp(1019)) in quantum physics, the large-k superoscillations in a function f(x) represent ‘weak values’ of momenta - values outside the spectrum of an operator, obtained in a ‘weak measurement’ - a gamma ray emerging from a box containing only red light

37. summary M V Berry 1987 ‘Quantum phase corrections from adiabatic iteration’ Proc. Roy. Soc. Lond. A 414, 31-46 M V Berry 2005 ‘Universal oscillations of high derivatives’ Proc. Roy. Soc. Lond. A 461, 1735-1751 David Farmer and Robert Rhoades 2005 ‘Differentiation evens out zero spacings’, Trans. Am. Math.Soc, S0002-9947(05)03721-9 Berry, M V, 1994, ﾔFaster than Fourierﾕ, in ﾔQuantum Coherence and Reality; in celebration of the 60th Birthday of Yakir Aharonovﾕﾊ(J S Anandan and J L Safko, eds.) World Scientific, Singapore, pp 55-65. Berry, M V, 1994, J.Phys.A 27, L391-L398, ﾔEvanescent and real waves in quantum billiards, and Gaussian Beamsﾕ. Berry, M V & Popescu, S, 2006, 'Evolution of quantum superoscillations, and optical superresolution without evanescent waves', J.Phys.A 39 6965-6977. asymptotically, functions can be dominated by their subdominant exponentials M V Berry,2004 ‘Asymptotic dominance by subdominant exponentials’, Proc. Roy. Soc. Lond, A,460, 2629-2636 under repeated differentiation, functions eventually oscillate universally functions can oscillate arbitrarily faster than their fastest fourier components