Vortices in Gaussian beams

Transcription

1 Vortices in Gaussian beams F. Stef Roux CSIR National Laser Centre PO Box 395, Pretoria 0001, South Africa CSIR National Laser Centre p.1/30

2 Contents Definition of polynomial Gaussian beam Calculation techniques and general properties of polynomial Gaussian beam Single vortex beams Gaussian beams with vortex dipoles CSIR National Laser Centre p.2/30

3 Gaussian beam notation Gaussian beam in normalised coordinates: g(u,v,t) = exp ( u2 + v 2 ) 1 it u = x ω 0 v = y ω 0 t = z ρ ρ = πω2 0 λ ω 0 1/e 2 beam waist radius; ρ Rayleigh range x Beam waist ω 0 ω(z) z ρ Rayleigh range ρ Rayleigh range CSIR National Laser Centre p.3/30

4 Gaussian beam Gaussian beam in terms of amplitude and phase g(u,v,t) = exp ( u2 + v 2 ) 1 + t 2 exp ( it(u2 + v 2 ) ) 1 + t 2 Normalised beam radius: 1 + t 2 Wavefront normalised radius of curvature: R(t) = 1 + t2 t CSIR National Laser Centre p.4/30

5 Polynomial prefactor Polynomial Gaussian beam = prefactor Gaussian beam g(u,v,t) Gaussian beam f(u,v,t) = P(u,v,t)g(u,v,t) P(u,v,t) complex bivariate polynomial (of finite order) "bivariate" means two variables (u and v). Coefficients depend on t. Bivariate polynomials cannot always be factorised. CSIR National Laser Centre p.5/30

6 Complex bivariate polynomials General form of complex bivariate polynomial prefactor: P(u,v,t) = N p α p,q (t)(u + iv) p q (u iv) q p=0 q=0 In terms of (u + iv) and (u iv) instead of u and v α p,q (t) complex coefficients as functions of the propagation distance t CSIR National Laser Centre p.6/30

7 Helical coordinates Occasionally it will be useful to re-express polynomial Gaussian beam in terms of helical coordinates: ( f(w,w,t) = P(w,w,t) exp ww ) 1 it where w = u + iv and w = u iv; and P(w,w,t) = N p α p,q (t)w p q w q p=0 q=0 CSIR National Laser Centre p.7/30

8 Complex bivariate polynomials p 4 (w,w) = α 00 + α 10 w + α 11 w + α 20 w 2 + α 21 ww + α 22 w 2 + α 30 w 3 + α 31 w 2 w + α 32 ww 2 + α 33 w 3 +α 40 w 4 + α 41 w 3 w + α 42 w 2 w 2 + α 43 ww 3 + α 44 w 4 Leading order by itself is fully factorisable in terms of product of 1st order polynomials. Each 1st order polynomial represents a complex zero an optical vortex. CSIR National Laser Centre p.8/30

9 Propagation We want to find out what the t-dependence is for a beam with a certain input function at t = t 0. Input function f(u,v) Gaussian beam t CSIR National Laser Centre p.9/30

10 Fresnel integral g(u,v,t) = ΩQ(u,v,t) f(u,v)q(u,v,t)k(u,v,t) dudv Ω t-dependent complex factor f(u,v) two-dimensional complex input function at t = t 0 [ Q(u,v,t) = exp i ( u 2 + v 2) ] t t 0 K(u,v,t) = exp [ i2 ( uu + vv ) ] t t 0 CSIR National Laser Centre p.10/30

11 Propagating Gaussian beams Fresnel propagation of polynomial Gaussian beam is represented by, g(u,v,t) = ΩQ(u,v,t)FR {P(u,v)g(u,v,t 0 )} P(u,v) polynomial prefactor in the input plane (at t = t 0 ) g(u,v,t 0 ) Gaussian beam in the input plane (at t = t 0 ) CSIR National Laser Centre p.11/30

12 Replacement Replace variable with derivative ( ) i2 u exp uu = t t 0 t t 0 i2 ( ) i2 u exp uu t t 0 Since the derivatives are independent of the integration variables they can be pulled out of the integral. CSIR National Laser Centre p.12/30

13 Operator approach One can simplify calculations, by turning polynomial prefactor into differential operator: FR {P(u,v)g(u,v,t 0 )} = P(d u,d v ) {FR {g(u,v,t 0 )}} Replacement: u d u = t t 0 i2 u v d v = t t 0 i2 v CSIR National Laser Centre p.13/30

14 Differentiation i.s.o integration Evaluate the integral over the Gaussian beam (once and for all). Then, instead of evaluating an integral every time, one only needs to perform differentiations: ( ) t0 + i g(u,v,t) = ΩQ(u,v,t) P (d u,d v ) {Γ(u,v,t)} t + i where we replace (u,v ) with (u,v). Γ(u,v,t) = exp [ i (t 0 + i)(u 2 + v 2 ] ) (t + i)(t t0) CSIR National Laser Centre p.14/30

15 Helical coordinates For propagation: g(w,w,t) = Ω exp ( iww ) ( ) t0 + i P (d w,d w ) {Γ(w,w,t)} t t 0 t + i where Γ(w,w,t) = exp [ ] (t 0 + i)ww i (t + i)(t t0) w d w = i(t t 0 ) w w d w = i(t t 0 ) w CSIR National Laser Centre p.15/30

16 Single noncanonical vortex Prefactor at t = t 0 : p 1 (w,w) = ξ(w w 0 ) + ζ(w w 0 ) where w 0 = u 0 + iv 0 and w 0 = u 0 iv 0. After propagation: ( p 1 (w,w,t) = ξ w [ t0 + i t + i ] w 0 ) + ζ ( w [ t0 + i t + i ] w 0 ) CSIR National Laser Centre p.16/30

17 Trajectories Steps to find the vortex trajectories: Substitute: w, w, ξ and ζ in terms of real and imaginary parts. Separate prefactor into real and imaginery parts. Find coordinates (u,v) as function of t where both parts become zero. CSIR National Laser Centre p.17/30

18 Trajectories for example Trajectories for p 1 = w w 0 (single off-axis canonical vortex). [ ] t0 + i p 1 = (u + iv) (u 0 + iv 0 ) t + i Trajectory: u(t) = u 0 v 0 t t 2 0 v(t) = u 0t 0 + v t u 0t 0 + v t 2 0 u 0 v 0 t t 2 0 t t CSIR National Laser Centre p.18/30

19 Visual example For v 0 = 0 (rotate beam) and t 0 = 0 (input in waist): u(t) = u 0 and v(t) = u 0 t The vortex propagate on a straight line lying in a plane parallel to axis. u u 0 t Beam waist Vortex trajectory v t CSIR National Laser Centre p.19/30

20 Vortex dipole input Input prefactor for vortex dipole (pair of oppositely charged vortices): p 2 (w,w) = [ξ 1 (w w 1 ) + ζ 1 (w w 1 )] [ξ 2 (w w 2 ) + ζ 2 (w w 2 )] w 1, w 1, w 2, w 2 locations of the two respective vortices ξ 1, ζ 1, ξ 2, ζ 2 morphologies of the two respective vortices CSIR National Laser Centre p.20/30

21 Vortex dipole propagated Propagation gives: ( [ t0 + i p 2 (w,w,t) = [ξ 1 w t + i ( [ξ 2 w [ t0 + i t + i (ζ 2 ξ 1 + ξ 2 ζ 1 )(t t 0 ) ] w 1 ) + ζ 1 ( w ] w 2 ) + ζ 2 ( w ( t0 + i t + i ) [ ] )] t0 + i w 1 t + i [ ] )] t0 + i w 2 t + i Result contains product of vortices, plus additional coupling term. Coupling term destroys factorisability CSIR National Laser Centre p.21/30

22 Annihilations Coupling term dipole annihilations and dipole creations. Without coupling term, prefactor is always factorisable fixed number of vortices. Points where dipole annihilations and dipole creations occur are called critical points. u t Dipole creation Beam waist Vortex trajectory Dipole annihilation v t CSIR National Laser Centre p.22/30

23 Simulated example CSIR National Laser Centre p.23/30

24 Dipole propagation example Follow same steps as before. Trajectory equations are in general second order in u and v only solutions when discriminant > 0 Example: two canonical vortices: p 2 = (w u 0 )(w + u 0 ) After propagation: p 2 (w,w,t) = [ ( ) t0 + i (u + iv) t + i [ ( ) t0 + i (u iv) t + i ( ) t0 + i i(t t 0 ) t + i ] u0 ] + u0 CSIR National Laser Centre p.24/30

25 Dipole trajectories Trajectory: v(t) = [ 1 u ] 0 2u t 2 (t t 0 ) 0 u(t) = ± 4 ( 1 + t 2 0) (1 + t 2 )u 4 0 (t t 0) 2 ( 1 + t 2 0 ) 2 2u2 0 2 ( 1 + t 2 0) u0 Part under square root must be positive. CSIR National Laser Centre p.25/30

26 Critical point analysis Critical points are where discriminant= 0. Find critical point by setting discriminant equal to 0 and solve for parameters u Parameter value (u 0 ) Critical value Threshold value Propagation distance (t) CSIR National Laser Centre p.26/30

27 Morphology evolusion The morphology for a vortex dipole changes during propagation. The compute the morphology: Compute trajectories for vortex dipole Compute morphology distributions Evaluate distributions at locations on trajectory CSIR National Laser Centre p.27/30

28 Morphology evolusion example B' B Anisotropy [π radians] F H D A E I G C C' Orientation [π radians] CSIR National Laser Centre p.28/30

29 Observation Vortices become edge dislocations (degenerate) near critical points. Vortices are only isotropic (canonical) in input plane. Morphology at t = is the same as the morphology at t =. CSIR National Laser Centre p.29/30

30 Conclusions Polynomial Gausian beams provide a relatively easy way to investigate vortex behaviour. Single vortices propagate in straight lines. Vortex dipoles give dipole creation and dipole annihilation. Vortex morphologies evolve for vortex dipoles. CSIR National Laser Centre p.30/30