Supplementary Information

Transcription

1 Suppleentary Inforation Curved singular beas for three-diensional particle anipulation Juanying Zhao 1-3 *, Ioannis Chreos 4,5 *, Daohong Song 1, Deetrios N. Christodoulides, Nikolaos K. Efreidis 4, and Zhigang Chen 1,3 1 The MOE Key Laboratory of Weak-Light Nonlinear Photonics, and TEDA Applied Physics Institute and School of Physics, Nankai University, Tianjin 3457, China CREOL/College of Optics, University of Central Florida, Orlando, Florida Departent of Physics and Astronoy, San Francisco State University, San Francisco, CA Departent of Matheatics and Applied Matheatics, University of Crete, Heraklion 7149, Greece 5 Max Planck Institute for the Science of Light, Erlangen D-9158, Gerany * These authors ade equal contribution. higang@sfsu.edu, nefre@uoc.edu S1: Media files The edia files contain two short ovies, illustrating the propagation dynaics of a selfaccelerating singular bea and a case of particle anipulation using a bea of this kind. The first ovie (Media 1 is an aniation of the propagation dynaics of an =1 Bessel-like vortex bea. Although the center is shifting and the ain lobe exhibits soe intensity rotation in the aiuthal direction, the sie of the ring reains constant as the bea propagates. The second ovie (Media shows the observed spinning of traped icroparticles in a transvsere plane by a triply-charged self-acceleating vortex bea propagating along a hyperbolic-secant trajectory. Note that the particles would undergo threediensional spiral otion should they have not been pushed against the holding glass. ( edia files attached

2 S: Detailed theoretical anaylsis of accelerating Bessel-like singular beas In the following we describe in detail the analysis leading to the design of accelerating singular beas of the higher-order Bessel type with arbitrary trajectories. Forulation of the proble We begin with the Fresnel integral of diffraction that describes the paraxial propagation of an optical wave Error! Reference source not found. ( x ξ + ( y η 1 uxy (,, = u, exp iq, + i d d π i + + ( ξ η ( ξ η ξ η (1 where the transverse coordinates x, y, ξ, η are noralied by an arbitrary length and the propagation distance by k (equivalent to the Rayleigh length, while k = π / λ is the wavenuber. The aplitude and phase of the wave at the input plane = is defined as (, (,,, e iq ξ η u ξη = u ( ( ξη The ain task of the analysis is to deterine the phase Q ( ξ, η that is required to produce an accelerating vortex bea with a given transverse width and a topological charge. As a easure of the transverse width of the bea, we use the diaeter of the inner low-intensity disk which is defined in Fig. S1. Figure S1: Real part of a vortex Bessel bea with order = 6. Indicated is the inner low-intensity disk which is defined at the radius where the arguent of the Bessel function is equal to its order. We now eploy ray optics. The equations of the rays follow fro the condition of first-order stationarity of the function P ( ξη, Q( ξη, ( x ξ + ( y η = + (3

3 which is the total phase of the wave coponent contributed to the field point ( x, y, by the input point ( ξη,, ( Pξ Pη = = we get. Setting the first-order partial derivatives of Eq.(3 equal to ero x ξ y η Qξ =, Qη = (4 which are the equations of a ray fro the input to the field point. Subsequently, we require that, at an arbitrary transverse plane, the rays eitted at skew angles with respect to the axis fro a (yet unknown locus L( on the input aperture pass fro a circle with center (,, = C f g and a fixed radius r. This circle we briefly denote as ( r C,. The subscript indicates a connection of this radius with the order of vorticity. The functions f, g( deterine the trajectory of the bea with the propagation distance acting as a paraeter. Using Eqs. (4, our requireent is expressed as where (, circle ( r Q ξ cos f + r ϕ ξ g + rsin, Q ϕ = η η = (5 ξη,, intersects the ϕ ξ η is the aiuth angle of the point at which the ray fro C. The reference direction, ϕ = for easuring the aiuth is arbitrarily taken to be the x-axis. At this point we also note that, if any input point ( ξη,, of its locus L( is apped to the distance, then a two-variable function (, is critical to finding Q ( ξ, η but is yet unknown. ξ η is obtained. The latter Deterination of the phase Q There are two key steps needed to proceed. The first is the requireent that the phase Q ( ξ, η and its first two derivatives are continuous. A necessary requireent is that its ixed second-order partial derivatives should be equal, i.e. Q ξη = Q, or using Eqs. (5 ( r cos ( r sin ( r sin ( r cos ξ ξ ϕ ϕ ϕ = η η ϕ + ϕ ϕ (6 η η ξ ξ where the subscripts ξ, η iply the partial derivatives of the corresponding functions and ξ f f η g g =, =. (7 ηξ

4 Equation (6 is a differential one for the unknown functions ( ξ, η and (, assue that the locus L( is a circle with center ( ξ, η, and radius all functions are to be deterined explicitly or equivalenty ( ξ ξ ( η η : ϕ ξ η. We now R, where L + = R (8 R cos, R ξ = ξ + θ η = η + sinθ (9 where θ is the aiuth coordinate of the point ( ξ, η on the circle L(. By differentiating Eq. (8 with respect to ξ and η we obtain the gradient ξ, η as ( cos,sin ξη, = ξη, cos θ,sin θ ( cos,sin θ θ θ θ = ξ cosθ + η sin θ + R D, θ (1 where the prie denotes the derivative d / d. Fro the obvious relation η η θ = arctan ξ ξ we also obtain the gradient ξ, ηθ as R ξη, θ ξη, ( R ξ sin θ η cos θ sin,cos = + θ θ (11 The gradient ξ, ηϕ can also be obtained if we require that angles θ and ϕ satisfy ϕ = θ + w (1 where w( is another function that is unknown for the oent. Fro the above equation we have ϕ = θ + w. (13 ξ, η ξ, η ξ, η Subsequently, we substitute Eqs. (9 into Eq. (6 and use Eqs. (1 to obtain:

5 r r ( cos sin sin ξ w+ η w u θ + ( ξ sin w η cos w + v cosθ R R (14 R 1 + r sin w + w cot w = R where ξ ξ η η u =, v = (15 Now notice that Eq. (14 is a trigonoetric series for θ which holds for all θ [,π if and only if all coefficients are ero. Hence R ξ cos w+ η sin w= u ( a r R ξ sin w η cos w= v ( b r R 1 + w cot w= ( c R (16 Equation (16 is easily integrated to find sin ( w β = (17 R where β > is a real integration constant. By solving Eqs. (16 for ξ and η we obtain the first-order ODE syste: ξ R cos w sin wu η = r sin w cos w v (18 Alternatively, using Eqs. (15 and (17, the syste becoes Rcos w r β u 1 u f v = r R cos w r + v g β (19 To solve this syste, the radius R( is required. This is found fro the field profile in the neighbourhood of the center C. Within ray optics, the field profile can be deterined approxiately by assuing that each ray eitted fro the circle L( contributes a plane

6 wave du in that region. Under the paraxial approxiation and neglecting the variations of their aplitude, these eleentary plane waves can be expressed as f + r cosϕ ξ ip( ξη, + i ( δx r cosϕ du (, θ exp dθ g+ r sinϕ η i ( δ y r sin ϕ + ( where ( x, y, ( x, y, ( f, g, observation ( x, y, fro C. δ δ = is the transverse displaceent of the point of Also, P( ξ, η is the value of the total phase fro the point ( ξ, η on the circle L( to the corresponding field point ( r, ϕ on the circle ( C, r. To find P( ξ, η, we differentiate Eq. P ( ξη, Q( ξη, = + ( x ξ + ( y η (3 with respect to ξ and η and use the syste (18 to find after soe long algebraic calculations where f ξ (, g η P ξη = r cos ϕ+ rsinϕ+ βrϕ+ W (1 r + R Rr w ζ ( 1 cos W = f + g u v dζ Now by inserting Eq. (9, (1 and (1 into Eq. ( we obtain r Rcos w f ξ g η du (, θ expiw + iβrw + ir + i δx + i δy r + R Rrcosw expiβ rθ + iρ cos( θ μ+ ν dθ (3

7 where δ x = ρcos μ, δ y ρsin μ = and ν polar coordinates around the center we finally get r sin arcsin w = r cos w R. Obviously ( ρμ, are C. Integrating all contributions over θ, i.e. du, π π u( ρθ,, πexpiw + iβrw+ ν+ ir r Rcos w f ξ g η r R Rrcosw + exp i δx i δy J βr ρ + exp( iβr μ (4 which reveals that the bea behaves locally around should we set C like a Bessel vortex of order β r, β r = (5 Notice that, in order to have vortex bea that resists diffraction, the factor ultiplying the polar radius ρ in the arguent of the Bessel function should be independent of, naely u J ( Mρ, or r + R Rr cosw = M (6 where the real constant M > represents the (noralied transverse wave nuber of the Bessel-like bea. Cobining the above with Eq. (17, we finally obtain the radius R = M + r + r M β (7 (Note that there is also a second solution R(, which is however rejected because it is not onotonic with the circles ust be expanding. At this point it would be interesting to coent that the result of Eq. (4 justifies our assuption of Eq. (8, naely, the rays are eitted fro circles on the input plane. This is a general property of Bessel-type paraxial waves: such waves result fro conical superpositions of rays eitted fro circles on the input apertures.

8 Now that the radius R( is known, functions w(, P( ξ, η, deterined fro Eqs. (17, (1 and (. The reaining functions, alternatively, W are readily ξ η or u v are deterined by solving the syste of Eq. (19. But before doing that, we need to relate the constants M and β. To this end, we note that the rays eitted fro a circle L( create an oblique conical-like surface with a nonero iniu waist (see Fig. S. The iniu waist ust be equal to r. To see this ore clearly, we express paraetrically a ray starting fro soe point on the circle L( with aiuth θ ( τ = ( 1 τ( ξ + Rcos θ, η + Rsin θ, + τ ( f + r cos ϕ, g+ r sin ϕ, r (8 where τ is a diensionless paraeter, with on the circle ( r, r being the starting point and r ( 1 the point C fro which the ray passes, according to Eq. (5. It is easy to see fro Eq. (8 that for any τ (hence at any propagation distance the rays pass fro a circle with center and radius ( τ = ( 1 τ( ξ, η, + τ (,, C f g (9 ( 1 cos cos (( 1 sin sin r τ = τ R θ + τr ϕ + τ R θ + τr ϕ ( cos τ ( cos = R + r Rr w R R r w τ + R (3 Using Eqs. (17 and (6 we easily get fro the above that r in = βr / M, which iplies that we ust have M = β (31 We now proceed to solve the syste (19, which, due to Eq. (31, siplifies to u 1 β u f = + v r β v g (3

9 y x Figure S: Scheatic of the principle. Rays eitted fro expanding circles on the input plane and at skew angles with respect to the axis interfere to create an oblique cone-like surface with a iniu waist diaeter equal to r (green circle. The field around the iniu waist is proportional to a vortex Bessel wavefunction of order. However, the initial condition that is required to solve this syste cannot be deterined fro Eq. (15 by siply letting ξ = η =, because we also have ξ η = =, thus getting an indefinite ratio /. For this reason we think as follows: the circles L( ust be expanding and never intersecting. This eans that the denoinator D(, θ in Eq. (1 ust be nonnegative for all θ which happens if and only if ξ η R R + = u + v (33 r for all, where we have used Eq. (18 to replace the functions ξ, η by u, v. Setting in the above u = and since R =, R + v naely u v r = fro Eq. (7 and (31, we get = =. Now the functions u, v can be copletely deterined fro the syste (3 once the trajectory functions f, g are given. Subsequently, the functions ξ, η are obtained fro Eqs. (15.

10 Algorith for coputing the phase Q Q ( ξ, η In the following, we outline the ain procedure for coputing the phase function. Given are the order of the singular bea, its iniu radius r (or alternatively its transverse wavenuber There are four steps: β = and the trajectory functions, / r f g. 1 For any point ( ξη,, on the input plane, solve Eq. (8 for to find the unique circle L( passing fro this point. Copute W( fro Eq. (, which now siplifies to where functions, ζ β ( W = f + g u v d ( u v have been deterined fro solving the syste of Eq. (3. 3 Copute P( ξ, η fro Eq. (1, where the phase ϕ is coputed fro Eq. (1 and w( η η fro Eq. (17, with θ = arctan. ξ ξ 4 Finally, obtain Q ( ξ, η fro Eq. P ( ξη, Q( ξη, = + ( x ξ + ( y η (3 as Q ( ξη, P( ξη, ( f + r cosϕ ξ + ( g + r sinϕ η = (35 Exaple: A vortex bea with constant acceleration As an exaple, let us consider the case of a singular bea with the D (lying on the x plane parabolic trajectory: f = γ, g( =. solution reads in ters of the center coordinates ξ, η as The syste (3 is exactly solvable and the γr β γr sin, β ξ = γ + η = 1 cos β r β r (36

11 y y (a (b x x Figure S3: (a Input phase, (b bea profile at = 1 and (c intensity cut on y = for the vortex bea with order = 4, diaeter r = 4 and parabolic trajectory (c x x= /4 (dotted curve. Inserting the above into Eq. (34, we also find exactly 3 3 4γ r β 4γ r β W = γ + + sin 3 3 β β r (37 Following the algorith described in the previous paragraph and using the last two equations, we are able to copute the input phase, noting that the only nuerical part of the procedure is the solution of Eq. (8 for through the Newton-Raphson ethod. An exaple is shown in Fig. S3 in ters of the input phase and intensity snapshots of the bea. The ray structure for this exaple is shown in Fig. S. A final note The condition Eq. (33 is a prerequisite for coputing the phase through the presented ethod, ensuring that the circles of Eq. (8 are expanding but never intersecting each other. However, for trajectories whose acceleration does not approach to ero as, this condition is satisfied only for distances below a certain bound, or ax. Beyond this distance, a new trajectory ust be defined in order to satisfy the condition, as for exaple a straight line. The procedure is then siilar to that of the ero-order Bessel beas discussed previously in Ref.

12 References [1] Goodan J., Introduction To Fourier Optics (Roberts and Copany Publishers, 5 [] Chreos I. D., Chen Z., Christodoulides D. N. & Efreidis N. K. Bessel-like optical beas with arbitrary trajectories. Optics Letters 37, 53-55, 1.