Generalization of helicoidal beams for short pulses

A generalization to the transient regime is developed for waves with a phase singularity of the screw type. These singular waves are commonly called vortices for all kind of waves as, for instance, optical vortex or acoustical vortex. We generalize the definition of vortices to get an azimuthal velocity invariant for all the frequency components contained in the broad spectrum of a short pulse. This generalization leads to a modification of the orbital angular momentum definition. Another generalization is introduced by considering helicoidal waves with a finite number of turns. We demonstrate that, in this last case, the topological charge is no longer an integer. This provides a physical interpretation to vortices of fractional charge that are involved here to take into account the diffraction occurring at both tips of the now finite helical wave front. We show that shortening the pulse implies an angular localization of the wave energy and, as a consequence, a spreading of the angular momentum amplitude due to the uncertainty principle.

Figure 1

(Color online) Helicoidal surface of pitch equal to 1 and truncated after three turns (left side), and of pitch equal to 3 truncated after one turn (right side).

Figure 2

(Color online) Sketch of the experimental setup. The measurements are made at $z=450$ and 600 mm whereas the fields are synthesized by inverse filtering at $z=450\text{mm}$.

Figure 3

(Color online) Wave front for a helicoidal beam with one turn (left), two turns (middle), and ten turns (right).

Figure 4

(Color online) (a),(c),(e) Pressure time signal and (b),(d),(f) frequency spectrum measured at $z=450\text{mm}$ for transient vortices of pitch $3T_0$ (blue dashed curves) and $4T_0$ (solid red curves) and for (a),(b) one; (c),(d) two; and (e),(f) ten helix turns.

Figure 5

(Color online) Pressure amplitude (gray scale in MPa) measured at $z=600\text{mm}$ for transient vortices of pitches (a) $3T_0$ and (b) $4T_0$ on a ring of width 0.5 mm and of radii 12 and 15 mm, respectively. The vortices are made of ten turns, but one located in the middle is presented.

Figure 6

(Color online) (a)–(c) Modulus (arbitrary units) and (d)–(f) phase $\left[-\pi ,+\pi \right]$ in gray scale. This subplot corresponds to the vortex of pitch $3T_0$, duration of ten turns, and measured at $z=600\text{mm}$. Three angular frequencies located on the three main spectral lines of Fig. 4, (a),(d) $2/3{\omega }_0$; (b),(e) $3/3{\omega }_0$; and (c),(f) $4/3{\omega }_0$, are selected.

Figure 7

(Color online) Phase measured at $z=600\text{mm}$ for transient vortices of pitch (a) $3T_0$ and (b) $4T_0$ on a ring of width 0.5 mm and of radii (a) 12 and (b) 15 mm, respectively. Three angular frequencies located on the three main spectral lines of Fig. 4: $\left(l-1\right)/l{\omega }_0$ (solid blue line), ${\omega }_0$ (dashed red line), and $\left(l+1\right)/l{\omega }_0$ (dashed-dotted green line).

Figure 8

(Color online) Pressure amplitude measured at $z=600\text{mm}$ for transient vortices of pitch (a) $3T_0$ and (b) $4T_0$ on a ring of width 0.5 mm and of radii 12 and 15 mm, respectively. The vortices are made of one turn.

Figure 9

(Color online) (a)–(c) Modulus and (d)–(f) phase of a vortex of pitch $3T_0$ and duration of one turn measured at $z=600\text{mm}$, at three angular frequencies located between two spectral lines of Fig. 4: (a),(d) $2.25/3{\omega }_0$; (b),(e) $2.5/3{\omega }_0$; and (c),(f) $2.75/3{\omega }_0$.

Figure 10

(Color online) Phase measured at $z=600\text{mm}$ for transient vortices of pitch $3T_0$ on a ring of width 0.5 mm and of radius 12 mm. Five angular frequencies located at $2/3{\omega }_0$ (solid blue line), $2.25/3{\omega }_0$ (dashed red line), $2.5/3{\omega }_0$ (dashed-dotted green bottom line), $2.75/3{\omega }_0$ (dotted cyan line), and ${\omega }_0$ (dashed-dotted black upper line) are plotted. The fitting lines obtained by linear regression are plotted with the same line types and colors.

Figure 11

Phase slope (dot), phase increment in multiple of $2\pi$ [Eq. (27) multiplied by $\omega$, circle], and average of the normalized angular momentum [Eq. (25) multiplied by $\omega$, cross].

Figure 12

(Color online) (a) Modulus and (b) phase of vortex of pitch $3T_0$ and duration of one turn measured at $z=450\text{mm}$. The angular frequency is located between two spectral lines of Fig. 4 at $2.5/3{\omega }_0$.