Title. Statistical behaviour of optical vortex fields. F. Stef Roux. CSIR National Laser Centre, South Africa

Transcription

1 . p.1/37 Title Statistical behaviour of optical vortex fields F. Stef Roux CSIR National Laser Centre, South Africa Colloquium presented at School of Physics National University of Ireland Galway, Ireland 21 September 2009

2 What are optical vortices?. p.2/37

3 . p.3/37 Topological charge V = x iy = ρexp( iφ) V + = x + iy = ρexp(iφ) y Contour: Unit circle -1 C +1 θ(x,y) ˆds = ν 2π Vortex x Vortex dipole = 2 oppositely charged vortices

4 . p.4/37 Topological charge conservation Vortices form lines in 3D annihilation and creation of vortex dipoles topological charge flow Topological charge is locally conserved. Net flow of topological charge into a closed volume is zero.

5 . p.5/37 Parameterisation Taylor series expansion around vortex at origin positive vortex negative vortex 0 + a x x + a y y +... = A exp(iω) [ξ(x + iy) + ζ(x iy)] complex coefficients morphology parameters overall amplitude and phase where ξ 2 + ζ 2 = 1

6 . p.6/37 Vortex shape For isotropic (canonical) vortices: ξ = 1 and ζ = 0 ν = +1 ξ = 0 and ζ = 1 ν = 1 All other cases are anisotropic (noncanonical) Canonical Noncanonical

7 . p.7/37 Vortex morphology Morphology of the vortex (anisotropy and orientation) is given by ξ and ζ. Analogues to Jones vectors (polarisation): η = [ ξ ζ ] = [ cos(ψ/2) exp(iβ/2) sin(ψ/2) exp( iβ/2) ] +1 Morphology angles: 0 ψ π helicity (cosψ) 0 β < 2π orientation Coordinates on a Bloch sphere Helicity Degenerate vortices Orientation -1 Canonical vortices

8 . p.8/37 Generating optical vortices Diffractive optical elements (DOEs) or Spatial light modulators (SLMs) For phase function (with vortices): θ(x, y) Amplitude function: t(x,y) = cos [θ(x,y)] Phase functions: t(x,y) = exp[iθ(x,y)]

9 Vortex density limitation The maximum topological charge density D in area with circumference L for wavelength λ: D L < 1 λ For spatial frequency > wavenumber purely evanescent waves exterior depletion. p.9/37

10 . p.10/37 Point vortex profile Phase only element point vortices (no amplitude modulation) Density limitation effective profile for point vortex (remove evanescent field)

11 . p.11/37 Scintillated optical beams Optical beam in a turbulent atmosphere: index variations cause random phase modulations leads to distortion of the optical beam. Weak scintillation continuous phase distortions that can be corrected by an adaptive optical system: Scintillated beam Corrected beam Beam splitter Wavefront sensor Control signal Deformable mirror

12 . p.12/37 Strong scintillation Strong scintillation optical vortices. conventional adaptive optics does not work anymore. Need to get rid of the vortices.

13 . p.13/37 Forced annihilation One idea to get rid of optical vortices in strongly scintillated optical beams is to force vortex dipoles to u annihilate sooner by introducing a special phase function. t Gaussian beam Vortex trajectory Dipole annihilation u t Phase function for forced annihilation

14 Annihilation in Gaussian beam The phase function behind an annihilation point looks similar to the continuous part of the phase function before the annihilation point. The latter somehow has the ability to cause annihilation. So one can use the former to force a vortex dipole to annihilate. u Dipole annihilation t Gaussian beam Vortex trajectory Before annihilation After annihilation. p.14/37

15 . p.15/37 Vortex removal procedure Procedure to remove optical vortices: Located all the optical vortices. Divide them into dipoles. Compute annihilation phase function for each dipole. Multiply beam with all these phase functions. Allow beam to propagate. Not extremely successfull! We need to understand the statistical behaviour of vortex fields better.

16 . p.16/37 Random vortex fields A random wave field (speckle field) = random vortex fields. f(x,y,z) = n α n exp(ik n x) k n propagation vectors inside cone angle θ α n random complex coefficients Amplitude Phase

17 . p.17/37 Properties of random vortex fields (... or what we already know) Vortex density inversely proportional to coherence area. Globally neutral topological charge. Adjacent topological charges are anti-correlated. Annihilation rate = creation rate System in equilibrium

18 . p.18/37 Perspective Physical beam: 2+1 dimensional world. Vortices = particles in 2D space. Propagation direction = time-dimension. Transverse plane (quasi) Random vortex field z Scintillated beam

19 . p.19/37 Vortex plasma model Vortex dipole annihilation Positive vortex (Hypothetical) neutral particle Negative vortex Vortex dipole creation Negative vortex Positive vortex Direction of propagation Three types of particles: Positive vortices n p (x,y,z) Negative vortices n n (x,y,z) Neutral bound states n 0 (x,y,z) (positive vortex + negative vortex)

20 . p.20/37 Conservation equations (1) (2) (3) z n p (x,y,z) + J p (x,y,z) = C A z n n (x,y,z) + J n (x,y,z) = C A z n 0 (x,y,z) + J 0 (x,y,z) = A C, J p (x,y,z), J n (x,y,z) and J 0 (x,y,z) currents associated with n p (x,y,z), n n (x,y,z) and n 0 (x,y,z), respectively. C rate of creations A rate of annihilations

21 . p.21/37 Vortex number conservation Total positive/negative vortex number density: Q P = n p + n 0 Q N = n n + n 0 Total positive/negative vortex number conservation: z Q P + J P = 0 where J P = J p + J 0 z Q N + J N = 0 where J N = J n + J 0 Total positive vortex number density: V = Q P + Q N = n p + n n + 2n 0 Total vortex number conservation: z V + J V = 0 where J V = J P + J N

22 . p.22/37 Disturbing the equilibrium Removing the continuous phase repeatedly Phase screens Scintillated beam Least squares phase removers Normalized total number of optical vortices Distances of free space propagation [A.U.] 1 km 10 km d System aperture Observation plane Initial number of optical vortices is reduced asymptotically until equilibrium is reached.

23 . p.23/37 Homogeneous case Remove continuous phase reduce n 0 (z) not in equilibrium Homogeneous + density limitation: globally neutral [n p (z) = n n (z)] Assume that: Rate of creations is proportional to n 0 (!) Rate of annihilations is proportional to n p For homogeneous densities: J = 0 Conservation equations become rate equation: z n p (z) = n 0(z) τ 0 z n 0 (z) = n p(z) τ p n p(z) τ p n 0(z) τ 0

24 Solution Conservation of positive vortex number: n N 0 n p (z) + n 0 (z) = Q P (z) = N 0 (constant) Solution: n p (z) = N 0τ p τ 0 + τ p + ( n p (0) N ) [ 0τ p exp (τ ] 0 + τ p )z τ 0 + τ p τ 0 τ p z If we assume: n 0 (0) = 0[ n p (0) = N 0 ] n p (z) = N 0τ p + N [ 0τ 0 exp (τ ] 0 + τ p )z τ 0 + τ p τ 0 + τ p τ 0 τ p Solution qualitatively agrees with observation, but since n 0 (0) is not known, one cannot solve for τ p and τ 0.. p.24/37

25 . p.25/37 Inhomogeneous case Numerical simulations for densities having regions with different topological charges Number of optical vortices Propagation distance [pixels] Vortex number increases during propagation. Restoration scale proportional to A in /λ

26 Why larger vortex density? Coherence area autocorrelation peak. Output vortex area [pixel-squared] Coherence area [pixel-squared] Separated topological charge: higher spatial frequencies narrower autocorrelation peak smaller coherence area larger vortex density. p.26/37

27 . p.27/37 Topological charge density Total vortex number depends on the hypothetical existence of n 0. To avoid n 0 consider the topological charge density: D = n p n n J D = J p J n Conservation of topological charge density: z D(x,y,z) + J D (x,y,z) = 0

28 . p.28/37 Diffusion equation Random motion of vortices random walk Diffusion equation: z D(x,z) κ 2 D(x,z) = 0 where κ is the diffusion coefficient and D(x,z) is the topological charge density (TCD) Solutions (exponential decay): D(x,z) = exp( κ a 2 z) cos(a x) Rate of decay increases for higher spatial frequencies.

29 . p.29/37 Numerical simulations (1D) Numerical simulation: Propagate initial beam cross-section: D(x,z = 0) = cos(ax) compute TCDs extract FT component Found Gaussian shape! broken shift invariance Modified equation: z D κ 0 z 2 D = 0 ( General solution: exp κ 0 2 a 2 z 2) cos(a x)

30 Phase drift Concentrated topological charge phase slope sideways drift x θ z k for k θ z x k z D(x,z + z) = D(x x,z) D(x,z) x D(x,z) for z 0: z D(x,z) = θ D(x,z) k. p.30/37

31 . p.31/37 Drift term Gradient of the phase function: θ = D(x,z) φ Drift term: z D(x,z) = 1 k [D(x,z) φ] D(x,z) where denotes convolution; and φ is the phase gradient of a single vortex: φ(x,y) = yˆx xŷ x 2 + y 2

32 . p.32/37 Full Fokker-Planck equation z D κ 0 z 2 D 1 k (D φ) D = 0 D topological charge density κ 0 dimensionless diffusion parameter k wavenumber φ phase function of a single canonical vortex

33 . p.33/37 Two-dimensional example Initial distribution: D(x,y,z) = cos (ax) + cos (by) Convolution: D φ = 2πŷ a sin (ax) + 2πˆx b φ(x,y) = sin (by) yˆx xŷ x 2 + y 2 Gradient: D = aˆx sin (ax) bŷ sin (by) Drift term: (D φ) D = 2π b2 a 2 ab sin (ax) sin (by) (D φ) D = 2π b2 a 2 [cos (ax by) cos (ax + by)] ab

34 . p.34/37 Nonlinearity The drift term only contributes for Fourier components with different magnitudes and different directions: cos(ax) and cos(by) New Fourier components: cos(ax + by) and cos(ax by) are generated due to the nonlinear drift term.

35 . p.35/37 Conclusions Strongly scintillated beams contain vortex fields that need to be removed to correct the beam. Special phase functions can force isolated dipoles to annihilate, but doesn t work well for vortex fields. The number of vortices are reduced by removing the continuous phase from homogeneous vortex fields. Vortex number increase during propagation of inhomogeneous vortex fields. A vortex plasma model is used to model the behaviour of vortex fields.

36 . p.36/37 More conclusions Vortex fields can be described by a Fokker-Planck equation that contains a drift term and a diffusion term. For one-dimensional distributions the drift term falls away and the result is a heat equation, but the decay follows a Gaussian shape because the system lacks shift invariance. For some two-dimensional distributions the drift term is nonzero and generates new Fourier components due to its nonlinear nature.

37 Mathematical Optics Team. p.37/37