Invisible Random Media And Diffraction Gratings That Don't Diffract

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$Invisible Random Media And Diffraction Gratings That Don't Diffract 29/08/2017 Christopher King, Simon Horsley and Tom Philbin,$

1 Invisible Random Media And Diffraction Gratings That Don't Diffract 29/08/2017 Christopher King, Simon Horsley and Tom Philbin, University of Exeter, United Kingdom, webpage:

2 Outline Aims, Motivation. Mathematically modelling wave propagation in inhomogeneous media. 1d reflectionless media: Perfectly absorbing layers. Perfectly transmitting disordered media. 2d non-scattering media Periodic media suppressing diffraction. Conclusions. 2

3 Aims and Motivation Reduce scattering from objects- invisibility. Possible using Transformation Optics [1,2] [3] [1]: J. B. Pendry, D. Schurig, D. R. Smith, Science , (2006). [2]: U. Leonhardt, Science , (2006). [3]: I. D. Rukhlenko, Plasmonics (2014). 3

4 Incident Reflected y 1d Graded index medium ε x x Helmholtz equation Transmitted TE polarized, monochromatic light of frequency ω. Isotropic, planar dielectric. μ = 1. d 2 dx 2 + k 0 2 ε x E x = 0 k 0 = ω/c Spatial Kramers-Kronig media are one-way reflectionless. [4] [4]: S. A. R. Horsley, C. G. King, T. G. Philbin, J. Opt , (2016). 4

5 Perfectly absorbing media (Complex-valued) ε x = 1 ln k 0x + 2i k 0 x + 2i E x, y E x, y y [5] x [5]: C. G. King, S. A. R. Horsley and T. G. Philbin, J. Opt , (2017). 5

6 An Experimental Realisation [9]: W. Jiang et. al., Laser Phot. Rev., , (2017). 6

7 Condition for real-valued ε(x) d 2 dx 2 + k 0 2 ε x E x = 0 k 0 = ω/c Ansatz: Modify a plane wave: E x = Ae ik 0x A(x)e iκs(x) amplitude A(x) phase S(x) both real ε is real if d dx ds A2 = 0 dx Solution E x = 1 x p( p(x) 1/4 eiκ x)d x [6] [6]: M. V. Berry, C. J. Howls, J. Phys. A 23 6, (1990). 7

8 Real-valued single frequency reflectionless media Corresponding permittivity ε x, κ = p x p(x)1/4 κ 2 d 2 dx 2 1 p(x) 1/4. [7,8] [7]: C. G. King, S. A. R. Horsley and T. G. Philbin, Phys. Rev. Let , (2017). [8]: K. G. Makris et. al., Light: Sci. Apps. 6, e17035, (2017). 8

9 y Incident plane wave x 2d Graded index medium Scattered waves ε x, y TE polarized, monochromatic light of frequency ω. Isotropic, planar dielectric. μ = 1. 2 x y 2 + k 0 2 ε x, y E x, y = 0 k 0 = ω/c 9

10 10 Condition for real-valued ε(x) Ansatz: Modify a plane wave: E x, y = Ae ik 0x A(x, y)e ik 0S(x,y) amplitude A(x, y) phase S(x, y) both real ε is real if. A 2 S = 0 Energy conservation: A 2 S is proportional to the time averaged Poynting vector (energy flux density).

11 The characteristic method. A 2 S = 0 (PDE) Choose S x, y. Find the exact rays. dλ x λ, y λ Solve for A along rays subject to boundary condition. Solve for ε. 11

$Non-diffracting$ periodic gratings ε

12 y x k 0 = 5 k 0 = Non-diffracting periodic gratings ε x, y E x, y k 0 =

13 13 Ongoing: Diffraction into a pair of modes A x, y y x

14 14 Conclusions 1d reflectionless perfectly absorbing media. 1d reflectionless perfectly transmitting disordered permittivity profiles. 2d Periodic non-diffracting media. Acknowledgements PhD supervisors: Dr Simon Horsley Dr Thomas Philbin EPSRC Centre for Doctoral Training in Electromagnetic Metamaterials EP/L015331/1.

15 15

16 16 Beamshifter Exact solution to. A 2 S = 0 ε x, y E x, y y x

17 17 Translationally Invariant rays. A 2 S = A A d2 S + 2 da ds + A d2 S + 2 da ds =0 dx 2 dx dx dy 2 dy dy 0 0 S x, y = s y + c x A x, y = 1 s y+c x c x ε x, y, k 0 = c x s y + c x k 0 2 g x, y

18 18 Translationally Invariant rays Phasefront ray slopes are independent ofy dy dx = 1 c x E.g. c x = sinh αx α

Isotropic, planar dielectric. x μ = 1. lim x ± ε x = 1.

19 Model: Graded index medium Incident Reflected ε x Transmitted y TE polarised, monochromatic light of frequency ω. Isotropic, planar dielectric. x μ = 1. lim x ± ε x = 1. d 2 dx 2 + k 0 2 ε x k y 2 E x = 0 k 0 = (k x, k y, 0) k 0 = ω/c Assume k y = 0 from now on. 19

20 A condition for no reflection (all angles, oneway incidence) Complex position: x z = x 1 + ix 2 (c.f. transformation optics). Kramers-Kronig media [1,2]: ε(z) analytic in the upper half position plane. lim ε z = 1 (Im z > 0). z x 2 x 2 αe k 0x 2 + βe ik 0x 2 γe k 0x 2 + δe ik 0x 2 ε z = 1 ε z = 1 Inhomogeneity αe ik 0x 1 + βe ik 0x 1 γe ik 0x 1 + δe ik 0x 1 Inhomogeneity x 1 [1]: S. A. R. Horsley, M. Artoni, G. C. La Rocca, Non reflecting permittivity profiles and the spatial Kramers Kronig relations, Nature Phot (2015). [2]: S. A. R. Horsley, C. G. King, T. G. Philbin, Wave propagation in complex coordinates, J. Opt (2016). 20

21 21 Transmission coefficient for Kramers-Kronig media Transmission t 2 = e 2k Im( 0 ε(z)dz) ε z = 1 + N k=1 a k z z k + N k=1 b k (z z k ) 2 +

22 22 Complex-valued Kramers-Kronig broadband reflectionless media Choose real-valued function with compact support Χ R x and ΧR x dx = 0. Spatial Hilbert transform Χ I x = 1 P Χ R x π x x satisfies ΧI x dx = 0. dx also ε x = 1 + Χ R x + iχ I x has transmission 1 for all incident frequencies and angles.

23 Hurst exponent, H: Measuring Randomness H > 0.5 implies long term positive correlations. H < 0.5 implies long term negative correlations. Χ x Χ Two-point correlation function g s = (x+s)dx. μ(g(s)) 0 as N. Χ x 2 dx Berry-Howls correlation Fourier series correlation H =0.18 H =0.53 (real part), 0.47 (imaginary part) 23

24 Incident Anderson (Strong) Localization N slabs of randomly chosen homogeneous lossless media: ε 1 ε 2 ε 3 ε 4 ε 5 Transmitted Reflected Logarithmic average of transmission coefficient t eff 2 = exp N 2 i=1 1 t i [4] [4]: M. V. Berry, S. Klein, Transparent mirrors: rays, waves and localization, Eur. J. Phys , (1996). 24

25 Measuring Randomness: Correlation function Two-point correlation function g s = Χ x Χ (x+s)dx where Χ x = ε x 1. Χ x 2 dx g(s) 1 Fourier series correlation Berry-Howls correlation 0.5 5λ 10λ 15λ μ(g(s)) 0 as N. 25

26 Phase as a harmonic function. A 2 S = A A 2 S + 2 A. S = S=0 implies S = Re h(z) where z = x + iy A. S = 0 implies A = A Im h y x ε z = dh dz 2 1 A Im h k 0 2 A Im h ε(h) 26

27 ε z = dh dz Take ε h = 1: Transformation Optics 2 ε z = dh dz ε(h) 2 z space Vacuum solutions in h space Reflectionless solutions in z space [4] e.g. h = z + a2 z [4]: U. Leonhardt, Optical Conformal Mapping, Science , (2006). h space 27

28 Transverse reflectionless profiles A Im h ε h = 1 k 2 0 A Im h Profiles homogeneous along direction of propagation. e.g. ε h = 1 + ε Im h 2 k 0 2 cosh 2 Im h Re E Im h Re h Im h 28

29 29 Condition for real-valued ε(x) d 2 dx 2 + k 0 2 ε x E x = 0 k 0 = ω/c Plane wave Ae ik 0x solves Helmholtz for ε x = 1. Ansatz: Modify a plane wave: E x = Ae ik 0x A(x)e iκs(x) amplitude A(x) phase S(x) both real

30 Condition for real-valued ε(x) ε x, κ = 1 d 2 A + ds κ 2 A dx 2 dx 2 + i 1 κa 2 d dx ds A2 dx real part imaginary part ε is real if d dx ds A2 = 0 dx Solution E x = 1 x p( p(x) 1/4 eiκ x)d x [3] p(x) real function with lim x ± p x = 1. [3]: M. V. Berry, C. J. Howls, Fake Airy functions and the asymptotics of reflectionlessness, J. Phys. A 23 6, (1990). 30

31 31 y Transmitted modes x period θ i

$y x θ i = 0 Non-diffracting$ 5 0 1.5 1.

32 y x θ i = 0 Non-diffracting periodic gratings- vary angle ε x, y E x, y θ i = π θ i = 2π

Summary of Beam Optics

Summary of Beam Optics Gaussian beams, waves with limited spatial extension perpendicular to propagation direction, Gaussian beam is solution of paraxial Helmholtz equation, Gaussian beam has parabolic