Optical singularity dynamics and spin-orbit interaction due to a normal-incident optical beam reflected at a plane dielectric interface

The degenerate case of normal incidence and reflection of an optical beam (both paraxial and nonparaxial) at a plane isotropic dielectric interface, which is azimuthally symmetric in terms of the momentum-spatial variation of Fresnel coefficients but not in terms of the fundamental polarization inhomogeneity of the incident field, requires in-depth analyses. In this paper, we use the reflection and transmission coefficient matrix formalism to derive an exact field expression of a normal-reflected diverging beam. The availability of the exact field information allows controlled variations of the system parameters, leading to significant dynamics of phase and polarization singularities hitherto unanticipated in the literature. We carry out a detailed exploration of these dynamics in our simulated system, and also verify them experimentally by using an appropriate setup. We then use Barnett's formalism to determine the associated orbital angular momentum (OAM) fluxes, leading to a subtle interpretation and mathematical characterization of spin-orbit interaction (SOI) in the system. Our paper thus represents a nontrivial unification of the most fundamental electromagnetic reflection and transmission problem at a plane dielectric interface and the emerging areas of optical singularity dynamics with their understanding in terms of OAM flux and SOI. The normal-incidence–retroreflection geometry being especially amenable to applications, these beam-field phenomena are anticipated to have applications in interface characterization, particle rotation and manipulation, and other nano-optical processes.

Figure 1

Example schemes of optical systems for beam reflection simulations and experiments. (a) A diverging beam with a nonzero central angle of incidence ${\theta }_{i0}$. (b) A normally incident focused beam (standard experimental setup). (c) A normally incident diverging beam and a virtual screen. Here, ${\mathbf{k}}_{i0}$ and ${\mathbf{k}}_{r0}$ are respectively the incident and reflected central wave vectors; the $z^{\left(S\right)}=0$ surface is the interface separating the dielectric media of refractive indices $n_1$ and $n_2$; and $S_R$ is the screen. In (b), ${\mathbf{k}}_{00}$ is the initial wave vector; $B_S$ is a beam splitter; and $L_N$ is a converging lens that focuses the beam at $z^{\left(S\right)}=0$.

Figure 2

The simulated optical system to analyze the normal incidence and reflection of a diverging optical beam (description in the text). The incident beam coordinate system $I(x^{\left(I\right)},y^{\left(I\right)},z^{\left(I\right)})$, the dielectric interface coordinate system $S(x^{\left(S\right)},y^{\left(S\right)},z^{\left(S\right)})$, and the reflected beam coordinate system $R(x,y,z)$ are related as follows: (1) $-{\widehat{\mathbf{z}}}^{\left(S\right)}$, is the outward normal to the interface; (2) ${\widehat{\mathbf{z}}}^{\left(I\right)}\parallel {\mathbf{k}}_{i0}$, the central incident wave-vector direction; (3) $\widehat{\mathbf{z}}\parallel {\mathbf{k}}_{r0}$, the central reflected wave-vector direction; (4) ${\widehat{\mathbf{y}}}^{\left(I\right)}={\widehat{\mathbf{y}}}^{\left(S\right)}=\widehat{\mathbf{y}}$; and (5) due to normal incidence, ${\widehat{\mathbf{x}}}^{\left(I\right)}={\widehat{\mathbf{x}}}^{\left(S\right)}=-\widehat{\mathbf{x}}$ and ${\widehat{\mathbf{z}}}^{\left(I\right)}={\widehat{\mathbf{z}}}^{\left(S\right)}=-\widehat{\mathbf{z}}$. For the simulation we have considered a virtual collimating lens $L_2$ and a virtual observation screen $S_R$.

Figure 3

The component field profiles ${\mathcal{E}}_x^X$, ${\mathcal{E}}_y^X$, ${\mathcal{E}}_x^Y$, and ${\mathcal{E}}_y^Y$ [Eqs. (3)] as functions of $(x,y)$ at the output beam cross section, for ${\theta }_E={45}^{\circ }$, arbitrary ${\mathrm{\Phi }}_E$, and the considered simulation parameters.

Figure 4

Profiles of the functions $a_{\pm }$, $b_{+}$ [Eqs. (7)], and ${\mathrm{\Phi }}_{+}$ [Eq. (8)] as functions of $(x,y)$ at the output beam cross section, for ${\theta }_E={45}^{\circ }$ and ${\mathrm{\Phi }}_E=+\pi /2$.

Figure 5

Profiles of the functions $a_{+}$, $b_{+}$ [Eq. (6b)], and ${\mathrm{\Phi }}_{+}$ [Eq. (8)] as functions of $(x,y)$ at the output beam cross section, for ${\mathrm{\Phi }}_E=+\pi /2$, (a–c) $\mathrm{\Delta }{\theta }_E=-0.{05}^{\circ }$, and (d–f) $\mathrm{\Delta }{\theta }_E=+0.{05}^{\circ }$ [70].

Figure 6

[(a)–(c)] Simulated intensity profiles ${\mathcal{I}}_{+}=(n_1/2{\mu }_{0}c){\left|{\mathcal{E}}_{+}\right|}^2$ for $\mathrm{\Delta }{\theta }_E=-0.{05}^{\circ },0^{\circ },+0.{05}^{\circ }$ and ${\mathrm{\Phi }}_E=+\pi /2$. [(d)–(f)] Corresponding total field profiles $\mathcal{E}$ [Eqs. (5) and (6)] [70]. The ellipticities of the polarization ellipses are not visually understandable, since the ${\widehat{\mathit{\sigma }}}^{-}$-polarized component field ${\mathcal{E}}_{-}$ significantly dominates over ${\mathcal{E}}_{+}$. However, the arrangement of the polarization ellipses is understood by generating the streamlines.

Figure 7

Variation of $d_0$ [Eq. (10c)] as a function of $\mathrm{\Delta }{\theta }_E$ in an example range $[-0.3^{\circ },+0.3^{\circ }]$, for the considered simulation parameters.

Figure 8

Trajectories of the singularity points—the $x$ and $y$ axes. The point pairs of the form $(\pm d_0,0)$ and $(0,\pm d_0)$ [Eqs. (10)], for all $d_0$ [Eq. (10c), Fig. 7], represent the positions of the singularity pairs for $\mathrm{\Delta }{\theta }_E\le 0$ and $\mathrm{\Delta }{\theta }_E\ge 0$ respectively. Representative points are shown on these trajectories, with $\mathrm{\Delta }{\theta }_E$ values marked. Here, any two adjacent points are separated by a $\mathrm{\Delta }{\theta }_E$ interval of $0.{025}^{\circ }$.

Figure 9

The experimental setup, comprising the following components. $L_S$, He-Ne laser; $L_P$, collimating lens pair; $H$, half wave plate (HWP); $G_1\text{and}{G}_S$, Glan-Thompson polarizers (GTP); $Q_1\text{and}{Q}_S$, quarter wave plates (QWP); $B_S$, beam splitter; $L_N$, high numerical aperture (NA) lens; $G_P$, glass plate (the $z^{\left(S\right)}=0$ surface is used as the isotropic dielectric interface for reflection); $C_C$, CCD camera (as screen $S_R$). The lens $L_N$ serves the purposes of both $L_1$ and $L_2$ of Fig. 2, and hence the coordinate systems $I(x^{\left(I\right)},y^{\left(I\right)},z^{\left(I\right)})$ and $R(x,y,z)$ are both defined with respect to the position of $L_N$.

Figure 10

[(a)–(c)] Experimentally obtained ${\mathcal{I}}_{+}$ intensity profiles for $\mathrm{\Delta }{\theta }_E=-1.{00}^{\circ },0.{00}^{\circ },+1.{00}^{\circ }$ and ${\mathrm{\Phi }}_E=+\pi /2$. [(d)–(f)] Corresponding single-slit diffraction patterns. The diffraction patterns exhibit appropriate fringe dislocations at the positions of the singularities [the length scales of (d), (e), and (f) are different from those of (a), (b), and (c), because the beam is expanded before passing through the single slit]. In (e), as we move along a central bright fringe towards the singularity, the fringe bends left and aligns with the first-order bright fringe on the other side of the singularity. This identifies a topological charge $t=-2$. In (d) and (f), as we move in a similar way, the central bright fringe bends left and aligns with the first dark fringe on the other side, and also partially connects to the first-order bright fringe. This identifies a topological charge $t=-1$. A technical aspect: In any directly displayed beam image, the $y$ pixel count starts at the top and proceeds downwards. But in a formal profile plot on the $xy$ plane, the $\widehat{\mathbf{y}}$ direction is vertically upwards. So the direct image is flipped vertically with respect to the true profile. The images of the single-slit diffraction patterns shown in Ref. [71] are thus flipped. The true patterns for $t=\pm 1$ are shown in Ref. [60], and are explained in terms of azimuthal energy flows in vortex beams.

Figure 11

Experimentally obtained representative points on the singularity trajectories, with $\mathrm{\Delta }{\theta }_E$ values indicated. Any two adjacent points are separated here by a $\mathrm{\Delta }{\theta }_E$ interval of $0.{25}^{\circ }$.

Figure 12

OAM flux density profiles $M_{\mathrm{orb}}^{\pm }$ [Eq. (14)] of the fields ${\mathcal{E}}_{\pm }$ [Eq. (6a)]. (a–c) $M_{\mathrm{orb}}^{-}$ for $\mathrm{\Delta }{\theta }_E=-0.{05}^{\circ },0^{\circ },+0.{05}^{\circ }$. (d–f) $M_{\mathrm{orb}}^{+}$ for $\mathrm{\Delta }{\theta }_E=-0.{05}^{\circ },0^{\circ },+0.{05}^{\circ }$ [70].

Figure 13

(a) Variations of the OAM fluxes $L_{\mathrm{orb}}^{\pm }$ [Eq. (15)] as functions of $\mathrm{\Delta }{\theta }_E$. Both the functions $L_{\mathrm{orb}}^{\pm }$ have identical variations around the central values (at $\mathrm{\Delta }{\theta }_E=0^{\circ }$) $L_{\mathrm{orb}}^{+\left(0\right)}$ and $0\hslash /s$. (b) Variations of the powers ${\mathcal{P}}_{\pm }$ [Eq. (17)] as functions of $\mathrm{\Delta }{\theta }_E$. The functions ${\mathcal{P}}_{\pm }$ have exact opposite variations around the central values ${\mathcal{P}}_{\pm }^{\left(0\right)}$—implying some energy transfer from the field ${\mathcal{E}}_{-}$ to the field ${\mathcal{E}}_{+}$ as $|\mathrm{\Delta }{\theta }_E|$ moves away from $0^{\circ }$, while satisfying energy conservation. This energy gain of ${\mathcal{E}}_{+}$ is consistent with the evolution of the ${\mathcal{I}}_{+}$ profiles of Fig. 6. (c) Variations of the “OAM per photon” $l_{\mathrm{orb}}^{\pm }$ [Eq. (18)] as functions of $\mathrm{\Delta }{\theta }_E$. The functions $l_{\mathrm{orb}}^{\pm }$ vary around the central values $-2\hslash$ and $0\hslash$. The variation of $l_{\mathrm{orb}}^{-}$ is represented here by multiplying it with a factor ${10}^{10}$.