Fig. 1. Exploration of conventional OAM beams versus IPVs. (a) Traditional OAM beams (${\mathrm{LG}}_{l,0}$) showcase OAM-dependent size and divergence, with each color representing a unique OAM order $l$. Conversely, IPVs manifest OAM-independent size and divergence (the orange curve), whereas $M_{\mathrm{IPV}}$ is directly derived from the square root of the selected $M_{\mathrm{IPV}}^2$. (b), (c) Complex field patterns for OAM beams (${\mathrm{LG}}_{l,0}$) and IPVs with identical beam waist but varying OAM orders. $M_{\mathrm{IPV}}^2$ and $M_{\mathrm{OAM}}^2$ are the quality factors of IPVs and OAM beams (${\mathrm{LG}}_{l,0}$). (d) Receivers with limited size obstruct the passage of OAM beams having large $l$ values due to increasing beam size and divergence as the mode index grows.¹⁴ However, (e) IPVs of any $l$ can easily traverse because of their OAM-independent propagation characteristics, maintaining their structure even after turbulence or obstacles.¹⁴

Fig. 2. Demonstration of OAM-dependent and OAM-independent propagation. The complex amplitude distributions of (a)–(c) “Perfect Laguerre–Gauss beams” (Ref. 19) and (d)–(f) “Perfect vortex beam” (Ref. 20) at $z=0$, 0.5, and 1 m, respectively. $w_0$, the beam waist at $z=0$. (g)–(i) The complex amplitude distributions of the innermost-ring-based IPVs with global $M_{\mathrm{IPV}}^2=3.7$ at $z=0$, $0.5z_0$, and $z_0$, respectively. (j)–(l) The corresponding experimental results for panels (g)–(i); especially, (j) is the interference patterns between the IPVs and a reference plane wave. The luminance and color of the color map refer to the intensity (Int) and phase, respectively; the green curves represent intensity profiles along the $x$ axis, and the horizontal orange dashed lines serve as a reference for indicating the size of the vortex rings. For further experimental details, refer to Sec. 5 of the Supplementary Material.

Fig. 3. Smaller quality factors and self-healing properties of IPVs. Quality factors of (a) LG beams-$M_{\mathrm{LG}}^2(l,p)$ and (c) $\mathrm{IPVs}\text{-}{M}_{\mathrm{IPV}}^2(l,p)$ for 10,000 lowest orders ($l$ and $p$ equal $\mathrm{0,1},\dots ,99$; results for $l<0$ are the same as those for $l>0$ and are omitted here). The corresponding distribution histograms are shown in panels (b) and (d). The IPV ($l=30$, $p=12$, $z_0=150\mathrm{mm}$) is blocked by a square obstacle at $z=-150\mathrm{mm}$: (e) experimental intensity maps at different $z$-axial locations (Video 1, MP4, 732 KB [URL: https://doi.org/10.1117/1.AP.6.3.036002.s1]); (f) transversal energy flow of panel (e), following from the cycle-average Poynting vector,²⁵ the red arrows indicate the value and direction of each flow (Video 2, MP4, 1.52 MB [URL: https://doi.org/10.1117/1.AP.6.3.036002.s2]), where PCC is PCC of innermost rings; panels (g) and (h) are the same as panels (e) and (f) but for OAM beams (i.e., ${\mathrm{LG}}_{l,0}$) (Video 3, MP4, 358 KB [URL: https://doi.org/10.1117/1.AP.6.3.036002.s3]; Video 4, MP4, 1.46 MB [URL: https://doi.org/10.1117/1.AP.6.3.036002.s4]). The sharp-edged square obstacle is produced as masks via the process of photoetching chrome patterns on a glass substrate. For further experimental details, refer to Sec. 5 of the Supplementary Material.

Fig. 4. Assessing free-space propagation amid atmospheric turbulence for the LG beam and corresponding innermost-ring-based IPV with $l=22$ and $p=5$ in panels (a)–(d) from 0 to 2000 m. The insets display the intensity patterns of the propagating LG beam at different distances, while the red circles represent the aperture to truncate the innermost ring. (e) The complex distributions for detected beams of IPVs with global $M_{\mathrm{IPV}}^2=5.6$ and the corresponding OAM beams at $z=1000\mathrm{m}$ against atmospheric turbulence; (f) the normalized intensity in detected modes for each launched mode in panel (e) at $z=1000\mathrm{m}$; (g)–(i) the cross talk matrices for LG beams, OAM beams, and IPVs.

Fig. 5. Image transmission by 24-bit IPV multiplexing with ultrahigh color fidelity. (a) True color image, “The Starry Night” by Vincent van Gogh (1889), with 24 bits of color depth and $2^{24}$ colors including $128\text{pixels}\times 128\text{pixels}$, and (b) three RGB layers of panel (a) were encoded from bit 1 to 24; (c) received true color image after recovering with an error rate of $2.45\times {10}^{-4}$. (d) Color distribution histograms of the true color image with 24 bits of color depth and the RGB layers with 8 bits of color depth. The red pixels indicate the incorrect data received. (e) Numbers $Q$ of independent spatial subchannels for spatial multiplexing techniques from $S=1$ to $S=30$; Emp., empirical. (f) The improvement of numbers of independent spatial subchannels in (e) versus $Q_{\mathrm{OAM}}$.