Figure 1

Method of construction, and experimental apparatus for generation and measurement of arbitrarily accelerating optical beams. (a) Ray trajectories. A plane wave (red arrows) passes through a phase mask $\phi (x_0)$ (green) located at $z=0$. The phase mask is obtained from Eqs. (5, 6), and the relation $x=f(z)$. Following Eq. (6), the rays of light are straight lines from the mask, which are tangent to $x=f(z)$. As an example, two rays are presented in black. The resulting intensity distribution is propagating along the predesigned trajectory $x=f(z)$ (blue curve). (b) Experimental apparatus (not to scale). A broad cw Gaussian beam (red arrows) is incident on a reflective spatial light modulator (SLM). The SLM (Hamamatsu LCOS-SLM) modulates the phase in the vertical ($x$) direction. The beam reflected from the SLM is deflected by a beam splitter (BS), and subsequently propagates in free space above an optical rail along the $z$ direction. A CCD camera slides along the rail and records the optical intensity in the $x\mathrm{\text{−}}y$ plane, for several $z$ planes along the rail. Since the images are invariant in $y$, we extract a single cross section along the $x$ direction from each image. Plotting these cross sections along the $z$ coordinates at which they were obtained, we construct an intensity cross-section in the $x\mathrm{\text{−}}z$ plane. Top Inset: Typical experimentally recorded $x\mathrm{\text{−}}y$ image at a given position $z$. Bottom Inset: Example of such an $x\mathrm{\text{−}}z$ cross section, whose peak intensity feature follows a curve of the form $x=a{z}^5$.Reuse & Permissions

Figure 2

Measured and simulated properties of arbitrarily accelerating optical beams. (a) Experimental (top) and simulated (bottom) optical intensity distributions for beams accelerating along the nonparabolic trajectories $x=a{z}^3$ and $x=b[e^{kz}-1]$. The green lines are the best fits of the positions of maximum intensity of the experimental data, to these trajectories. These lines are redrawn on the simulated intensities, highlighting the agreement between theory and experiment. (b) Summary of a set of experiments of the type presented in (a) for predesigned polynomial trajectories $x=a_n{z}^n$ intersecting at $z=150\mathrm{cm}$. The data set for each beam is presented in a different color. The whole lines are the predesigned trajectories used for calculating the masks providing the initial field phases. The “+” marks are the positions of the peaks of the experimentally-obtained intensity distribution, at each plane $z$. The dashed lines are best fits of these data to the predesigned trajectories, the same as the green lines in (a). The data are plotted on a logarithmic scale, where the different accelerations of the beams appear as different linear slopes. (c) The ${\mathrm{Ai}}^2$-shaped intensity pattern transverse to the propagation axis. Experimentally obtained cross sections along the $x$ axis for the intensity distribution following the trajectories $x=a_5{z}^5$ (blue) and $x=a_3{z}^3$ (green), are compared to an ${\mathrm{Ai}}^2(x)$ function (red), in accordance with Eq. (4). (d) Experimental (whole lines) and simulated (dashed lines) full width at half maximum (FWHM) of the main-intensity feature, for beams which accelerate along the curves $x=a_5{z}^5$ (blue [dark gray]) and $x=b[e^{kz}-1]$ (green [medium gray]). The peak-intensity is nonbroadening, and even narrows up to $\sim 30$ Rayleigh lengths of propagation, after which the beam abruptly starts to broaden.Reuse & Permissions