Universal form of the carrier frequency of scalar and vector paraxial $X$ waves with orbital angular momentum and arbitrary frequency spectrum

We report on the dependence of the carrier frequency of a nondiffracting optical pulse on the amount of orbital angular momentum it carries. We provide a unified universal form of such a dependence for the cases of both scalar and vector pulses with arbitrary frequency spectra. For the case of paraxial optical pulses we consider two different examples, namely, pulses with exponentially decaying spectra and Gaussian spectra.

Figure 1

Dependence of the carrier frequency ${\omega }_c$ of a scalar (blue line) and a vector (red dashed line) pulse with an exponentially decaying spectrum from the OAM parameter $m$. As can be seen, ${\omega }_c$ varies linearly with $m$ in both cases. The scalar (or vector) nature of the pulse only accounts for a different value of ${\omega }_c$ at $m=0$. For this graph, $\alpha =1$ is assumed, meaning that ${\omega }_c$ is a dimensionless quantity.

Figure 2

Dependence of the carrier frequency ${\omega }_c$ of a scalar (blue line) and a vector (red dashed line) pulse with a Gaussian spectrum from the OAM parameter $m$. As can be seen, the dependence of ${\omega }_c$ on $m$ is more complicated that the case shown in Fig. 1, as in this case a Gaussian spectrum is assumed. The scalar (or vector) nature of the pulse only accounts for a different value of ${\omega }_c$ at $m=0$. For this graph, $\alpha =1$ is assumed, meaning that ${\omega }_c$ is a dimensionless quantity.

Figure 3

Plot of the intensity distribution of the components of the electric field as given by Eqs. (B3) in the plane $\zeta =0$: (a) $|E_x{(\mathbf{r},t)|}^2$, (b) $|E_y{(\mathbf{r},t)|}^2$, and (c) $|E_z{(\mathbf{r},t)|}^2$. The $x$ and $y$ axes are expressed in units of ${\vartheta }_0$. For these plots, the following parameters were used: $m=1, {\vartheta }_0=1$, and $\alpha =1$.

Figure 4

Plot of the intensity distribution of the components of the magnetic field as given by Eqs. (B5) in the plane $\zeta =0$: (a) $|B_x{(\mathbf{r},t)|}^2$ and (b) $|B_y{(\mathbf{r},t)|}^2$. The $x$ and $y$ axes are expressed in units of ${\vartheta }_0$. For these plots, the following parameters were used: $m=1, {\vartheta }_0=1$, and $\alpha =1$.

Figure 5

Plot of the intensity distribution of the components of the electric field as given by Eqs. (C6) in the plane $\zeta =0$: (a) $|E_x{(\mathbf{r},t)|}^2$, (b) $|E_y{(\mathbf{r},t)|}^2$, and (c) $|E_z{(\mathbf{r},t)|}^2$. The $x$ and $y$ axes are expressed in units of ${\vartheta }_0$. For these plots, the following parameters were used: $m=1, {\vartheta }_0=1$, and $\alpha =1$.

Figure 6

Plot of the intensity distribution of the components of the magnetic field as given by Eqs. (C9) in the plane $\zeta =0$: (a) $|B_x{(\mathbf{r},t)|}^2$ and (b) $|B_y{(\mathbf{r},t)|}^2$. The $x$ and $y$ axes are expressed in units of ${\vartheta }_0$. For these plots, the following parameters were used: $m=1, {\vartheta }_0=1$, and $\alpha =1$.