Fig. 1. Snapshot of the amplitude and phase of the scalar electric field u = ∣u∣ exp(iarg(u)) in a transversal plane through the waist of a Laguerre-Gaussian mode with radial index p = 0 and azimuthal index ℓ = 1 (donut mode). The axis of propagation z is perpendicular to the plane of the drawing, and passes through the center at (x,y) = (0,0). The brightness of the picture indicates the magnitude-squared amplitude ∣u∣² of the field; the color indicates the phase angle arg(u) = -ℓφ according to the color strip on the right [φ = arctan(y/x) is the azimuthal angle in the plane of the drawing].

Fig. 2. Computer-generated binary patterns based on Eq. (1), for K = 2π. Each pattern consists of 1000 × 1000 pixels, with pixels for which T = 0 rendered black and T = 1, white. Pattern (a) has no encoded vortex (M = 0 in Eq. (1)). Pattern (b) has a single fringe bifurcation (M = 1); when used as a hologram it gives rise to vortices with topological charge ±1 in the ±first diffraction order.

Fig. 3. Schematic of our 2f - 2f setup (not to scale). Ultrashort pulses enter from the left, and then pass through the following optical elements: G₁ = line grating without vortex fingerprint (see inset); A = order-selecting aperture; L₁ = planoconvex lens with focal length f; G₂ = grating with vortex fingerprint (see inset); L₂ = same as L₁. Diffraction orders (-1, 0, +1) are indicated as black numbers on a yellow background. The positions labeled with boxed lowercase letters (a … g) in light gray are used in our wavefront calculations (see Sec. 3). A few colored rays are shown to remind the reader that there is spatial chirp; white solid arrows indicate spatial chirp is absent. The artist’s impressions on the right suggest the resulting far-field radiation: the +1 diffraction order from G₂ gives a spatial-chirp-free donut mode (white ring), while the -1 diffraction order suffers from spatial chirp. Note that we show just three colors-in reality, the frequency spectrum is of course continuous.

Fig. 4. Images of optical vortices. Intensities in arbitrary units as indicated by the color bar in the center (negative values resulting from background subtraction are indicated in gray). Top row: laser free-running (narrowband radiation). Bottom row: laser mode-locked (femtosecond pulses, broadband radiation). Left column: no compensation of spatial chirp (diffraction order N = -1). Right column: spatial chirp compensated (diffraction order N = +1). The line graphs above and to the left of each image show the intensity distributions (in arbitrary units) along the horizontal and vertical dotted lines. Note the clean dark center in panel (d), and its absence in panel (c) due to spatial chirp. See Sec. 4 for more details.

Fig. 5. Interferogram of an ultrashort optical vortex. To obtain this image, we let the chirp-free optical vortex-containing donut mode as shown in Fig. 4(d) interfere with a vortex-free spherical reference wave. This reference wave was created by allowing the zero-order radiation coming from grating G₁ (see Fig. 3) to propagate through the setup. To get good fringe visibility, we attenuated this reference wave by using a tiny pinhole (see text). A one-armed spiral can be recognized in the image, indicating that the topological charge of the vortex in our ultrashort pulse equals 1.