Supporting Information: Ultraintense. femtosecond magnetic nanoprobes induced by. azimuthally polarized laser beams

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1 Supporting Information: Ultraintense femtosecond magnetic nanoprobes induced by azimuthally polarized laser beams Manuel Blanco, Ferran Cambronero, M. Teresa Flores-Arias, Enrique Conejero Jarque, Luis Plaja, and Carlos Hernández-García, Grupo de investigación Photonics4Life, Área de Óptica, Departamento de Física Aplicada, Universidade de Santiago de Compostela, Santiago de Compostela, E-15782, Spain Grupo de Investigación en Aplicaciones del Láser y Fotónica, Departamento de Física Aplicada, University of Salamanca, E-37008, Salamanca, Spain carloshergar@usal.es Abstract In this Supplemental Material we show: (I) the PIC simulations parameters and the spatial and temporal profile of the radial component of the magnetic field carried by an azimuthally polarized laser field, with and without the metallic aperture; (II) the temporal evolution of the spatial profiles of the vector beam E and B fields propagating through different apertures; (III) a sensitivity analysis of the B field amplification with the aperture shape; and (IV) a theoretical analysis of the enhanced longitudinal magnetic field. 1

2 I. PIC simulations parameters and Radial magnetic field In this work we compute the electric (E) and magnetic (B) fields through numerical simulations using the OSIRIS particle-in-cell (PIC) code. 1 3 PIC codes are an useful tool to calculate the interaction between laser fields and charged particles, solving in a regular grid Maxwell s and Lorentz s equations. These codes solve in a regular grid Maxwell s and Lorentz s equations to update the electromagnetic fields and particle dynamics, respectively. This is done periodically in a loop, such that the new positions and velocities of the particles computed at each time step can be used to obtain the charge and current densities, which allow to update the electromagnetic fields and restart the loop. In our 3D PIC simulations, the metallic aperture contains electrons and protons with a number density of n = 34.5n c where n c = meω2 4πq 2 e is the critical plasma density and ω and q e are the plasma frequency and the charge respectively. The target has a thickness of 50 nm, and the number of particles per cell is per specie. The density has a steep profile. The simulation box has a y and z axes of µ and x-axis of 1.65 µm. The spatial resolution is 5 nm in both axes. The simulation advances in time steps of 335 attoseconds. The energy conservation has been verified in all the simulations. The magnetic field (B) carried by an azimuthally polarized laser beam presents radial and longitudinal components, as stated in Eq. (2) in the main text. In Figs. 1 and 2 also in the main text we presented the azimuthal component of the electric field (E ϕ ) and the longitudinal of the magnetic field (B x ) without and with the circular aperture, respectively. In addition to those fields, the radial component of the magnetic field (B ρ ) is present in our configuration. In Fig. S1 we show the B ρ field without (left column) and with the aperture of ρ A = 1.78 µm (right column). The first row shows the longitudinal profile plane XZ at y = 0, whereas the second row shows the transverse profile at the focal plane x = 0 for the nonapertured case, and at the sample plane (x = λ/2) for the apertured case. In Figs. S1(e) and S1(f) we present the temporal profiles of B ρ for the non-apertured and the apertured 2

3 Figure S1: Bρ carried by an azimuthally polarized beam, without (left column) and with (right column) a circular aperture of radius ρa = ρ0 = 1.78 µm placed at x = 0 (black dashed line). (a) and (b) show the longitudinal spatial profiles of Bρ at y = 0. In the second row we show the transverse profiles (c) at the focal plane x = 0 without the aperture and (d) at a sample plane at x = λ/2 = 0.4 µm (black solid line) including the aperture. The corresponding temporal profiles of Bρ are shown at (e) ρ = ρ0 and x = 0 and at (f) ρ = 0.24 µm and x = λ/2. In the later case the incident field is also shown (dashed line). case, respectively. In the later case, we compare the incident field (without the aperture, dashed line), and the Bρ field a the sample, in the spatial position of maximum amplitude. In a similar way as the Eϕ, we observe that the Bρ is focused, but is not enhanced by the aperture. Thus, the use of the metallic aperture also helps to enhance the Bx field with respect to the Bρ field. The ratio Bx /Bρ as a function of the radial position (ρ) was shown in Fig. 4d of the main text. 3

4 II. Temporal evolution of the apertured azimuthally polarized field In order to show the physics behind the enhancement of the Bx field when using the metallic aperture, we first show in this section the temporal evolution of the electromagnetic field carried by the vector beam when using an aperture of ρa = ρ0. The incident driving pulse has a temporal duration of 4.8 fs full width at half maximum in intensity. Figure S2: Temporal evolution of the spatial Eϕ (left column) and Bx (right column) fields when using a gold circular aperture with ρa = 1.78 µm placed at x = 0. The different panels show the longitudinal plane XZ, y = 0 profiles at 4.6 fs (first row), 9.9 fs (second row), 14.3 fs (third row) and 17.9 fs (fourth row). Note that incident driving pulse exhibits a temporal duration of 4.8 fs full width at half maximum in intensity. 4

5 In Fig. S2 we show the longitudinal profile XZ plane at y = 0 of E ϕ (left column) and B x (right column), for which the aperture is placed at x = 0. The different rows show the profiles at 4.6 fs, 9.9 fs, 14.3 fs and 17.9 fs. Remarkably, one can observe how the aperture induces a strong B x field at its edge (Fig. S2d), which results in a delayed enhanced B x field on axis (ρ = 0). In the attached movies ( EphiXZapertured.avi and BxXZapertured.avi ) we present the temporal evolution of the E ϕ and B x fields respectively, for four different aperture radii (ρ A ): 0.25ρ 0, ρ 0 (the optimal one), 1.5ρ 0 and 3.4ρ 0. The later case could be considered as a non-apertured case. It is interesting to observe how the B x field reaches its maximum case for ρ A = ρ 0, whereas the E ϕ is not substantially enhanced when using the aperture. It can also be observed how the enhanced B x field is temporally delayed with respect to the driving field. In order to analyze in more detail the temporal structure of the B x field, we present in Fig. S3 the absolute value of the B x field as a function of time for four different aperture radii: (a) 0.25ρ 0, (b) ρ 0, (c) 1.5ρ 0 and (d) 3.4ρ 0. First we observe that the maximum B x field is reached when using the optimal aperture (ρ A = ρ 0 ). Second, as observed in the attached movies, the maximum enhanced B x field is delayed with respect to that carried by the incident azimuthally polarized laser beam, which is very similar to that shown in Fig. S3d (where the effect of the aperture is negligible). Thus, the choice of the aperture radii allows us not only to select the maximum amplitude of the B x field, but also its temporal envelope and pulse duration. Finally, in the attached movie ( BxXZthickness.avi ) we present the temporal evolution of the B x field respectively, for four different aperture thicknesses: 0.1µm, 0.4µm, 0.8µm and 1.2µm. Note that the thickness used in the rest of this work is 50 nm. While the amplification of the B x field remains similar, the spatial shape of the nanoprobe varies, favoring forward propagation against backwards propagation. 5

6 Figure S3: Temporal profile of B x at ρ = 0 for different aperture radii (ρ A ): (a) 0.44 µm (0.25ρ 0 ), (b) 1.78 µm (ρ 0 ), (c) 2.65 µm (1.5ρ 0 ), and (d) 6 µm (3.4ρ 0 ). In the later case (d) the effect of the aperture is negligible, an thus it mainly shows the B x field carried by the azimuthal laser beam. In the other panels, where the aperture has a noticeable effect over the electromagnetic field, the B x field is modified, being maximum in panel (b), where ρ A = ρ 0. We also observe that the enhanced B x field is delayed with respect to the incident B x field. III. Sensitivity analysis of the aperture shape Up to now we have considered a perfect circular aperture. In order to predict the effect of possible deviations from the circular aperture due to manufacturing defects on the B field amplification, we have performed PIC simulations considering elliptical apertures. In Fig. S4 we show the amplification of the B x field for an elliptical aperture versus the ellipticity of the aperture (ε A ). We define ε A as the ratio between the major axis (b) and the minor axis of the aperture ellipse (ρ A ), where the latter is fixed to the optimal aperture radius (ρ A = ρ 0 ), thus ε A = ρ A /b (see right inset in Fig. S4). The amplification ratio, B x /Bx, 0 is presented for different aperture ellipticities (ε A ), where B x is measured at the sample (x = λ/2), and 6

7 Bx 0 is the incident field without the aperture. Note that the case of ε A = 1 corresponds to the use of a circular aperture of radius ρ A = ρ 0, thus recovering the amplification factor of 6 obtained in the main text. It can be clearly observed that the B x field amplification is optimal for a circular aperture, being reduced when decreasing the aperture ellipticity. We note that the amplification reduces to 50% of its optimal value when the aperture presents an ellipticity of ε A = We attribute this reduction to the distortion of the azimuthal electric currents in the metallic aperture due to its elliptical shape. In addition, we note that for ε A < 0.6 the B x field amplification reaches a constant behavior with ε A, as the major axis of the aperture ellipse exceeds substantially the radius of maximum intensity of the driving beam, i.e. b >> ρ 0. Figure S4: Amplification of the B x field for an elliptical aperture versus the ellipticity of the aperture (ε A ). The right inset shows how we define the ellipticity of the aperture, by the ratio between the major axis (b) and the minor axis of the aperture ellipse, where the latter is fixed to the optimal aperture radius (ρ A = ρ 0 ). The ratio B x /B 0 x at ρ = 0 is presented for different aperture ellipticities (ε A ), where B x is measured at the sample (x = λ/2), and B 0 x is the incident field without the aperture. The rest of parameters are the same as those used in the previous sections. Note that the case of ε A = 1 corresponds to the use of an optimal circular aperture of radius ρ A = ρ 0. It can be observed that the B x field amplification is optimal for a circular aperture, being reduced when decreasing the aperture ellipticity. 7

8 IV. Theoretical analysis of the enhanced longitudinal magnetic field In this section we develop a model analysis of the properties of the magnetic field produced by the vector beam. Considering a linear isotropic response, the electric field of the azimuthally-polarized vector beam induces an azimuthal ring-shaped circular harmonic current distribution, j(ρ, φ, t) = j(ρ)e iωt e φ (S1) where ω is the frequency of the electric field. For simplicity, we shall consider a thin plate, located at x = 0, therefore the potential vector induced at any point in space is given by the retarded integral 4 A(r, t) = 1 c ( 1 r r e iω t r r c ) j (ρ, φ ) e φ ρ dρ dφ (S2) where we use the primed spatial coordinates to define the source points (the plane at x = 0). IV.1. Thin annular current distribution Let us first assume the radial current distribution as thin loop, so that the radial current density can be approximated by a singularity, j (ρ ) q t δ(ρ ρ 0 ) with q t being the total charge. Substituting into (S2), we have the following formula for the azimuthal component of the vector potential A φ (ρ 0 ; r, t) = q t e ik r r 0 c e iωt r r 0 cos φ dφ (S3) with k = ω/c, and r 0 the position of the source points at the circle of radius ρ 0. In cylindrical coordinates r r 0 = (ρ 2 + ρ 2 0 2ρρ 0 cosφ + z 2 ) 1/2. On the other hand, the radial component A ρ vanishes. Correspondingly, the longitudinal component of the magnetic field is given by 8

9 ( A) x = A φ /ρ + A φ / ρ. Near the x axis, the magnetic field can be approximated as B x,0 = lim ρ 0 ( A) x = 2 ( A φ / ρ) ρ=0. Using (S3), we find an expression for the complex magnetic field amplitude B x,0 = B x,0 e iϕ 0, B x,0 ( ρ 0 ; x) 1 ( ) ρ 2 1/2 0 4π 2 1 (S4) λ ρ x 2 ρ x 2 ( ) ϕ 0 ( ρ 0 ; x, t) = 2π ρ x 2 arctan 2π ρ x 2 ωt (S5) where we have defined the normalized spatial coordinates ρ 0 = ρ 0 /λ and x = x/λ. We show in Fig. S5a the longitudinal magnetic field amplitudes along the propagation axis, for two different instants of the oscillation, and the corresponding amplitude profile B x. The results for the time dependent amplitudes are obtained from the real part of the rotational of the vector potential, from the numerical integration of Eq. (S3) and the amplitude profile is obtained from Eq. (S4). In Fig. S5b we compare the profile obtained in our model with the exact results for our PIC calculations (Fig. 2e in the main text). The main characteristics are well-reproduced by the model, specially at the coordinate x 2λ that, at this time instant, corresponds to the field induced by the central part of the pulse considered in the exact calculations. Our monochromatic model fails to describe the decrease of the field amplitude at both sides of this central part, as it reflects the modulation of the few-cycle pulse envelope used in the exact calculation. IV.2. Thick annular current distribution Let us now consider description, closer to the situation exposed in the main text: a finite size annular current distribution, limited its inner radius by the aperture of the plate, with radius ρ A. The current density is assumed as a radial Gaussian distribution of half-width σ, centered at ρ 0. The total field corresponds to the weighted superposition of the elemental field amplitudes, corresponding to thin distributions of the kind considered in the previous subsection, Eqs. (S4) and (S5), of different radius. Therefore, near the longitudinal axis, we 9

10 a) ( ) / b) /λ ( ) Figure S5: Time-dependent amplitudes and amplitude profile of the longitudinal component of the magnetic field along the aperture axis, as a function of the distance to the current loop plane. (a) The blue curve represents the amplitude at the time instant when it is maximum at x = 0, and the orange curve a quarter of cycle later. Both cases are computed from the rotational of the numerical integration of the vector potential, from the numerical integration of Eq. (S3). The amplitude profile is shown in green, and corresponds to the modulus of the complex field, as given by Eq. (S4). (b) comparison of the amplitude profiles described by the model with the obtained by the exact PIC integration (Fig. 3a in the main text). The exact profile shows an amplitude modulation, caused by the field s pulse envelope, which is not present in our monochromatic model. /λ have (ρ ρ0)2 /σ2 B A (ρ A, ρ 0, σ; x) = 2π e B x,0 (ρ; x)ρdρ ρ A (S6) We can use Eq. (S6) to explore the choice of geometry to optimize the energy of the magnetic field induced at the aperture. To do this, we compute numerically the integral (S6) using Eqs. (S4) and (S5) at x = 0. The graphic shown in Fig. S6 corresponds to the aperture radius, ρ A, with the width of the annular current, σ, for which the magnetic field energy at the aperture is maximal, calculated for different wavelengths. As a general trend, 10

11 the induced magnetic field is maximized for aperture sizes of radii ρ A comparable to that of the annular current distribution, ρ 0. The departures from this value are more pronounced for longer wavelengths (i.e. comparable with the size of the current distribution). In all cases, it is found that the magnetic field energy is optimized at ρ A = ρ 0, when the aperture diameter is equal to one wavelength. ρ /ρ λ= ρ λ= ρ λ=ρ λ= ρ Figure S6: Aperture radius (ρ A ) maximizing the energy of the magnetic field induced at the aperture, as a function of the annular width of the current distribution (σ), for four different wavelengths of the driving field. The geometrical parameters are normalized to the radius of the central part of the current distribution, ρ 0. σ/ρ References (1) Fonseca, R. A.; Silva, L. O.; Tsung, F. S.; Decyk, V. K.; Lu, W.; Ren, C.; Mori, W. B.; Deng, S.; Lee, S.; Katsouleas, T.; Adam, J. C. OSIRIS: A Three-Dimensional, Fully Relativistic Particle in Cell Code for Modeling Plasma Based Accelerators. Computational Science ICCS Berlin, Heidelberg, 2002; pp (2) Fonseca, R. A.; Martins, S. F.; Silva, L. O.; Tonge, J. W.; Tsung, F. S.; Mori, W. B. Oneto-one direct modeling of experiments and astrophysical scenarios: pushing the envelope on kinetic plasma simulations. Plasma Physics and Controlled Fusion 2008, 50,

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