Figure 1

RCP Gaussian beam propagating along the negative $z$ axis; ${\lambda }_{\mathrm{o}}$ $=$ 0.5 μm; FWHM $=$ 4 μm. Shown in the cross-sectional $xy$ plane are the amplitude and phase profiles of $E_x$, $E_y$, $E_z$, and also the Poynting vector components $S_z$ and ${\mathit{S}}_{\perp }$. The phase distributions of $E_x$ and $E_y$ are uniform: ϕ($E_x$) $=$ 0°, ϕ($E_y$) $=$ 90°. $E_z$ is relatively weak, but its phase profile shows a 2π vorticity around the $z$ axis. The clockwise circulation of ${\mathit{S}}_{\perp }$ around $z$ is responsible for the SAM of the beam.Reuse & Permissions

Figure 2

RCP Gaussian beam arrives at the surface of a PEC mirror at the oblique incidence angle θ; the incident AM is denoted by ${\mathit{J}}_{\mathrm{inc}}$. Upon reflection, the $s$ component of the incident $E$ field reverses direction, while the $p$ component reorients itself to remain perpendicular to the propagation direction. The projections on the mirror surface of both ${\mathit{E}}_p$ and ${\mathit{E}}_s$ of the incident and reflected beams cancel each other out at $z$ $=$ 0. The reflected beam is thus left-circularly polarized, having the same AM content as the incident beam (i.e., $J_{\mathrm{ref}}$ $=$ $J_{\mathrm{inc}}$), but oriented differently. While the $z$ components of ${\mathit{J}}_{\mathrm{inc}}$ and ${\mathit{J}}_{\mathrm{ref}}$ are identical, their $x$ components are equal and opposite. This change of the $x$ component of AM upon reflection produces a net torque on the mirror, which tends to rotate it around the $x$ axis.Reuse & Permissions

Figure 3

Wedge-shaped reflector consists of two PEC sheets joined at 90°. The length, width, and height of the reflector used in the simulations are $l$ $=$ $w$ $=$ 12 μm, $h$ $=$ 6 μm. The RCP Gaussian beam of Fig. 1 is incident from above. After successive reflections at the sheets, the beam returns along the positive $z$ axis, having retained its RCP state of polarization.Reuse & Permissions

Figure 4

Cross-sectional amplitude and phase profiles of $E_x$, $E_y$, $E_z$ for the Gaussian beam of Fig. 1, after reflection from the 90° PEC wedge depicted in Fig. 3. The phase profiles of $E_x$ and $E_y$ have acquired some curvature, but ϕ($E_x$)−ϕ($E_y$) ≈ 90°. The $z$ component has retained its 2π vorticity, albeit with reversed handedness. The counterclockwise circulation of ${\mathit{S}}_{\perp }$ around $z$ confirms the reversal of the AM direction upon reflection from the wedge.Reuse & Permissions

Figure 5

(a) Hollow PEC conical reflector having a 90° apex angle. In our simulations, the cone's base radius and height were $R$ $=$ $h$ $=$ 6 μm. The RCP Gaussian beam of Fig. 1 is incident from above, along the negative $z$ axis. After two successive reflections at the conical surface, the beam returns along $+$$z$ with the same polarization state as the incident beam (RCP), but also having acquired vorticity with a 4π phase winding. (b) Cross-sectional view of the cone as seen from above. The ray arriving at time $t$ $=$ 0 at point $A$ on the cone surface is reflected twice before emerging at A′. For this ray, the $x$ and $y$ components of the $E$ field coincide with $p$- and $s$-polarization directions, respectively. Ignoring the constant time delay needed for all such rays to traverse a given cross-sectional plane, the ray entering at $A$ will have the same phase as the ray emerging at A′. The two reflections are responsible for $E_y^{\prime }$ being parallel to $E_y$, and $E_x^{\prime }$ being antiparallel to $E_x$. For the incident ray at $A$, the sense of polarization is right-circular, meaning that the $E$ field rotates from $E_x$ toward $E_y$. Similarly, the sense of rotation for the reflected ray emerging at A′ (also right-circular) is from $E_x^{\prime }$ toward $E_y^{\prime }$. With regard to the incident ray at $B$, since at $t$ $=$ 0 the $E$ field is aligned with the $x$ axis, it takes a certain fraction of the oscillation period for E to arrive at the local $p$ direction. When this ray crosses the cone and arrives at B′, its $E$ field must rotate still further to come into alignment with $E_x^{\prime }$. Therefore, the phase difference between the rays emerging at A′ and B′ is twice the angle θ between the $x$ axis and the local $p$ direction at $B$. This explains the appearance of a 4π phase winding on the reflected beam that emerges from the cone.Reuse & Permissions

Figure 6

Reflection of the RCP Gaussian beam of Fig. 1 from the hollow PEC cone depicted in Fig. 5a. The reflected beam is also RCP, as can be seen in the plot of ϕ($E_x$)−ϕ($E_y$), which is nearly constant at ∼90° (modulo 2π). Both ϕ($E_x$) and ϕ($E_y$) exhibit 4π vorticity around the $z$ axis; the phase of $E_z$, however, exhibits only a 2π spiral. The doughnut-shaped profiles of $E_x$, $E_y$, $S_z$ hint at the vorticity of the reflected beam. The clockwise circulation of ${\mathit{S}}_{\perp }$ around $z$ is dominated by vorticity, which opposes the counterclockwise circulation due to the beam's polarization state. The overall AM of the beam remains the same before and after reflection from the PEC cone.Reuse & Permissions

Figure 7

Parabolic PEC mirror brings a collimated incident beam propagating along the negative $z$ axis to diffraction-limited focus at its focal point $F$. The height of the simulated mirror is $h$ $=$ 3 μm, its radius at the top is $R$ $=$ 6 μm, and its focal plane is located 3 μm above the $xy$ plane.Reuse & Permissions

Figure 8

Various properties of the reflected beam at the focal plane of the paraboloid depicted in Fig. 7; the incident beam is that shown in Fig. 1. Whereas $E_x$ and $E_y$ are vortex free, there is a 2π vorticity in the $E_z$ component of the reflected beam. While the transverse component ${\mathit{S}}_{\perp }$ of the Poynting vector contains contributions from both SAM and OAM, it is difficult to isolate their individual contributions in these pictures. The integral of $S_z$ over the $xy$ plane yields the total optical power of the reflected beam, which turns out to be equal to that of the incident beam, as expected. The integral of r × ${\mathit{S}}_{\perp }$/$c^2$ over the focal plane, however, overestimates the total AM of the beam by as much as 44%; this is due to the substantial departure from paraxiality of the focused beam.Reuse & Permissions

Figure 9

Transparent dielectric cone having a 90° apex angle and an antireflection coating layer on the top facet. In our simulations, the base radius and the height were $R$ $=$ $h$ $=$ 6 μm, the refractive index of the coating layer was the square root of the cone's index, and the coating layer's thickness was ${\lambda }_{\mathrm{o}}$/(4$n_{\mathrm{coat}}$). The RCP Gaussian beam of Fig. 1 is incident from above, along the negative $z$ axis. The beam returns along the positive $z$ axis after two successive total internal reflections at the conical surface.Reuse & Permissions

Figure 10

Reflection of the RCP Gaussian beam of Fig. 1 from a glass cone ($n_{\mathrm{cone}}$ $=$ 1.55, $R$ $=$ 6 μm, $h$ $=$ 6 μm, $n_{\mathrm{coat}}$ $=$ 1.245, $t_{\mathrm{coat}}$ $=$ 100 nm). For the chosen $n_{\mathrm{cone}}$, the phase difference imparted to the light's $s$ and $p$ components upon each TIR is 45°. The reflected beam is thus linearly polarized, as can be seen in the plot of ϕ($E_x$)−ϕ($E_y$), which is nearly zero in some regions and almost 180° in others. Both $E_x$ and $E_y$ show 4π vorticity in their phase structure, but the corresponding OAM is only half as much as that of a full 4π vortex, because |$E_x$| and |$E_y$| are not uniformly distributed around $z$. The $z$ component of the field exhibits only a 2π vorticity. The plot of $S_z$ is doughnutlike, albeit with a partially filled hole. The clockwise circulation of ${\mathit{S}}_{\perp }$ around $z$ is almost entirely associated with the OAM of the (linearly polarized) reflected beam. The proximity of the 45° angle of incidence on the conical surface to the critical TIR angle of 40.18° is responsible for a small fraction of the incident light leaking out of the cone. In our simulation, the actual fraction of the incident light that returned along $+$$z$ was 92.1%; the corresponding AM of the returning beam was 92.6% that of the incident.Reuse & Permissions

Figure 11

Reflection of the RCP Gaussian beam of Fig. 1 from a glass cone ($n_{\mathrm{cone}}$ $=$ 2.56, $R$ $=$ 6 μm, $h$ $=$ 6 μm, $n_{\mathrm{coat}}$ $=$ 1.6, $t_{\mathrm{coat}}$ $=$ 78 nm). For the chosen $n_{\mathrm{cone}}$, the phase difference imparted to the light's $s$ and $p$ components upon each TIR is 79.3°. The reflected beam is thus elliptically polarized with a fixed degree of ellipticity but varying orientations of the polarization ellipse. This may be seen in the plots of |$E_x$|, |$E_y$|, and ϕ($E_x$)−ϕ($E_y$), with the last showing as much as 43° phase variation over the beam's cross section. Both $E_x$ and $E_y$ show slight variations in their phase profiles over the beam's cross section, but no multiple-of-2π vorticity, although $E_z$ exhibits a 2π vortex. The plots of $S_z$ and ${\mathit{S}}_{\perp }$ no longer possess circular symmetry around $z$, which is probably an artifact caused by insufficient numerical accuracy. The clockwise circulation of ${\mathit{S}}_{\perp }$ around $z$ is characteristic of the beam's total AM, which contains a mixture of SAM and OAM. In our simulation, the fraction of the incident light that returned along the positive $z$ axis was nearly 100%; the corresponding AM of the returning beam was also about 100% that of the incident. The beam's AM is thus fully conserved upon reflection, although a fraction of its SAM has been converted to OAM.Reuse & Permissions