Effect of Orbital Angular Momentum on Nondiffracting Ultrashort Optical Pulses

We introduce a new class of nondiffracting optical pulses possessing orbital angular momentum. By generalizing the $X$-wave solution of the Maxwell equation, we discover the coupling between angular momentum and the temporal degrees of freedom of ultrashort pulses. The spatial twist of propagation invariant light pulse turns out to be directly related to the number of optical cycles. Our results may trigger the development of novel multilevel classical and quantum transmission channels free of dispersion and diffraction. They may also find application in the manipulation of nanostructured objects by ultrashort pulses and for novel approaches to the spatiotemporal measurements in ultrafast photonics.

Figure 1

Real (a) and imaginary (b) part of the electric field component $E_x(\mathbf{r},t)$ as a function of the comoving coordinate $\zeta =(z/c)\cos {\vartheta }_0-t$, for the case $m=1$. The real part is an odd function that crosses the line $\zeta =0$ three times, while the imaginary part is an even function, with two crossings. Parameters: ${\vartheta }_0=0.01$ (corresponding to the paraxial case) and $\alpha =1$; $\zeta$ is dimensionless.

Figure 2

(a) Real part of the $x$ component of the electric field $E_x(\mathbf{r},t)$ for different values of the OAM parameter $m$. (b) Oscillatory function $\cos \psi (\mathbf{r},\zeta )$ for different values of the OAM parameter. The choice of these values corresponds to ${\omega }_c=3/\alpha$, ${\omega }_c=6/\alpha =2({\omega }_c){|}_{m=1}$, and ${\omega }_c=9/\alpha =3({\omega }_c){|}_{m=1}$, respectively. When the OAM increases, the carrier frequency blueshifts, with an increase of the number of optical cycles, but the envelope FWHM decreases, thus resulting in a shorter pulse. (c) Calculated (red dots) number of oscillations performed by $\cos \psi (\mathbf{r},\zeta )$ as a function of $m$. With every three units of $m$ added, the number of cycles of $\cos \psi (\mathbf{r},\zeta )$ grows by one unity. The number of oscillations is calculated by counting the number of zero crossing $N_0$ of each cosine term and using $N_0=2N_{\text{osc}}+1$. Here, ${\vartheta }_0=0.01$ (corresponding to the paraxial case) and $\alpha =1$ has been chosen. In particular, $\alpha =1$ means that $\zeta$ is a dimensionless quantity.

Figure 3

(a) Pulse envelope of the $x$ component of the electric field $E_x(\mathbf{r},t)$ for different values of the OAM parameter $m$. (b) Calculated (red dots) and fitted (blue dashed line) pulse width $\mathrm{\Delta }\tau$ as a function of the OAM $m$. The fit predicts an exponential shortening of the pulse width with respect to $m$, according to $A{e}^{-Bm}+C$. We have $A=1.16$, $B=0.28$, and $C=0.64$. Here, ${\vartheta }_0=0.01$ (corresponding to the paraxial case) and $\alpha =1$ has been chosen. In particular, $\alpha =1$ means that $\zeta$ is a dimensionless quantity.