Localized waves with spherical harmonic symmetries

We introduce a class of propagation invariant spatiotemporal optical wave packets with spherical harmonic symmetries in their field configurations. The evolution of these light orbitals is considered theoretically in anomalously dispersive media, and their spinning dynamics are analyzed in terms of their corresponding energy flows. Similarly, localized waves generated via spherical superposition from Archimedean and Platonic solids in $\vec{k}-\omega$ space are investigated in this work.

Figure 1

(a) Intensity isosurface plots of an $\ell =m=0$ localized wave. (b) Intensity cross section reveals the $j_0\left(R\right)\propto \sin R/R$ profile of this “$s$” state.

Figure 2

(a) Intensity isosurface plots of an $\ell =1,m=0$ localized wave. (b) Intensity cross section of “$p$” shell for $X=0$.

Figure 3

Intensity isosurfaces corresponding to higher order localized waves having $m=0$ when (a) $\ell =2$, (b) $\ell =3$, (c) $\ell =4$, and (d) $\ell =7$.

Figure 4

(a) Intensity isosurfaces of an $\ell =2$; $m=\pm 2$ localized wave. (b) Top view of power circulation when $m=+2$. (c) Power circulation in this same state when $m=-2$.

Figure 5

Superimposing localized waves with $\ell =2$; $m=\pm m_0$. (a) Intensity isosurfaces with $m_0=1$. (b) Isosurfaces with $m_0=2$.

Figure 6

A rotating localized wave.

Figure 7

Propagation dynamics of an apodized localized wave with $\ell =2;m=0$ after a normalized distance of (a) $Z=0$, (b) $Z=2.66$, and (c) $Z=4$. Values are normalized to the maximum value of the wave which occurs at $Z=0$.

Figure 8

The vertices of a (a) regular hexahedron, (b) octahedron, and (c) dodecahedron inscribed in a $Q$ sphere. (d)–(f) The corresponding isointensity patterns generated from these arrangements.

Figure 9

Isosurface spectrum plots of the $\ell =m=0$ localized wave with various degrees of Gaussian apodization. The spherical spectrum has been sectioned in half so that the shell thickness can be viewed. (a) $w=5$. (b) $w=10$. (c) $w=20$. In the limit that $w\to \infty$, the shell thickness becomes infinitesimally small, representing the spectrum of the $O$ wave.

Figure 10

Isosurface spectrum plots of the $\ell =1,m=0$ localized wave with various degrees of Gaussian apodization. The spherical spectrum has been sectioned in half so that the shell thickness can be viewed. (a) $w=5$. (b) $w=10$. (c) $w=20$.