Скачать презентацию Three universal oscillatory asymptotic phenomena Michael Berry Physics Скачать презентацию Three universal oscillatory asymptotic phenomena Michael Berry Physics

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

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

F plane stokes line dominant antistokes line subdominant F plane stokes line dominant antistokes line subdominant

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

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

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

phase contours of g 1(F) phase contours of g 1(F)

Re. F DSE Im. F Re. F subdominant DSE Im. F dominant Re. F DSE Im. F Re. F subdominant DSE Im. F dominant

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

application of f 4(z) to conical diffraction emerging from the slab, an axial spike application of f 4(z) to conical diffraction emerging from the slab, an axial spike light incident on a slab of crystal with three different principal dielectric constants crystal slab and a cylinder of light, radius R 0 Hamilton, Lloyd, Poggenforff, Raman… focal image plane entrance face wavelength l M V Berry 2004 ‘Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike’, J. Opt. A 6, 289 -300

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

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

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

if z(t) analytic in a strip including the real axis: singularities (poles or branch if z(t) analytic in a strip including the real axis: singularities (poles or branch points) closest to real axis Imz t 1++it 2+ Rez t t 1 --it 2 - C

assume for simplicity that t 2+≠t 2 - (e. g. z(t) not real), and assume for simplicity that t 2+≠t 2 - (e. g. z(t) not real), and define t 2 =min(t 2±) then, up to a constant and (real) shift of t origin, 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,

the nth loop has n/2+1 windings Imz Rez z 0 z 1 z 2 the nth loop has n/2+1 windings Imz Rez z 0 z 1 z 2 z 9 z 500 universal attractor of hodograph map (‘velocity’ dz/dt)

as n increases, “universal cosine” emerges near t=0: Rez 0(t) Rez 2(t) Rez 10(t) as n increases, “universal cosine” emerges near t=0: Rez 0(t) Rez 2(t) Rez 10(t) Rez 50(t) Rez 100(t) Rez 500(t) instability of differentiation?

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

geometric phase corrections are determined by time-dependent transformation to eigenbasis of H( t), giving geometric phase corrections are determined by time-dependent transformation to eigenbasis of H( t), giving H 1(t), and then iterating to H 2(t), etc; phase corrections are related to areas of Hn loops in the simplest nontrivial case, iterated loops are given by the derivative (hodograph) map: phase corrections get smaller and then inevitably diverge, universally

if z(t) has no finite singularities, asymptotics of zn(t) determined by saddle-points in integral 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…

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

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

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

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

deportment of saddle us( , 3) Imus incre asing Reus small , us=i. A deportment of saddle us( , 3) Imus incre asing Reus small , us=i. A large x, us 0 superoscillations normal oscillations, exponentially larger

local wavenumber kloc( ) cosh 2 A=2 local wavenumber kloc( ) cosh 2 A=2

A=4, =0. 2 log 10 Ref x x superoscillations, period sech 4=0. 037 x A=4, =0. 2 log 10 Ref x x superoscillations, period sech 4=0. 037 x normal oscillations, period 1

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

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

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

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

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

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

generalization: integrate over a to generate arbitrary functions locally, as band-limited function example: a 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

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

in signal processing, superoscillations in a function f(t) emerging from a perfect low-pass filter 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 9 th symphony - a one-hour signal, requiring up to 20 k. Hz for accurate reproduction - can be generated by a 1 Hz 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

summary asymptotically, functions can be dominated by their subdominant exponentials M V Berry, 2004 summary 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 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, S 0002 -9947(05)03721 -9 functions can oscillate arbitrarily faster than their fastest fourier components Berry, M V, 1994, ヤFaster than Fourierユ, in ヤQuantum Coherence and Reality; in celebration of the 60 th 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, L 391 -L 398, 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.