Anatomy of the geometric phase for an optical vortex transiting a lens

We present an analytical means of quantifying the fractional accumulation of the geometric phase for an optical vortex transiting a cylindrical lens. The standard fiber bundle of a sphere of modes is endowed with a supplementary product space at each point so that the beam waists and their positions can be explicitly tracked as functions of lens transit fraction. The method is applied to quantify the accumulation of the geometric phase across a single lens as a function of initial state and lens position within the beam. It can be readily applied to a series of lenses as well.

Figure 1

Fiber space. The sphere of modes (brown) is the projection of a 3D product space (pink) $\{w_1,w_2,z_2\}$ onto its associated fiber (green) followed by a projection of the fiber position $\chi$ down to its base. Subscripts 1 and 2 refer to principle axes of an elliptical beam cross section. Here $z=0$ is defined as the position of waist $w_1$

Figure 2

Lens transit. A thin converging cylindrical lens changes the waist and waist position along its active axis. Subscripts $p$ and $a$ refer to the passive and active axes of the cylindrical lens.

Figure 3

Sphere of Modes trajectories for lenses 1 and 2. A Gaussian beam (7) with spinor state (8) and SOM angles of $\varphi =\pi /2$ and $\theta =\pi /4$ (black sphere, $\zeta =0$) transits two individual cylindrical lenses in separate experiments. Their focal lengths are described by Eq. (15a) with ${\alpha }_1=\pi /2$ and ${\alpha }_2=\pi /4$. As a reminder, $\alpha$ is the difference in Gouy phase between active and passive axes after propagation through the lens, the difference that changes the mode of the beam. The spinor evolves according to Eq. (20), with $f$ replaced by $f/\zeta$, generating the blue curve shown. The length of this trajectory depends on lens position $z_{\ell }=z_{\text{rel}}{d}_{\text{nat}}$, with $d_{\text{nat}}$ given by Eq. (15b) for each lens. In all cases, the end point of an arc is associated with $\zeta =1$.

Figure 4

Arc length of Sphere of Modes trajectories. As shown in Fig. 3, the complete trajectory arc length is a function of lens position $z_{\ell }$. The total angle subtended is given by Eq. (24) and is plotted here for lenses 1 and 2. The dashed line is at an arc angle of ${90}^{\circ }$ and the associated lens positions are $z_{\ell 1}=0.59z_{Rp}$ and $z_{\ell 2}=1.77z_{Rp}$. These correspond to $z_{\text{rel},1}=1.41$ and $z_{\text{rel},2}=2.61$.

Figure 5

Trajectory through the Supplementary Product Space. Figures 3 and 4 show the sphere of modes trajectories associated with beam transit through two lenses; here the corresponding trajectories along fibers and through the supplementary product space are plotted using Eqs. (14b), (14c), and (25). The lens positions $z_{\text{rel},1}=1.41$ and $z_{\text{rel},2}=2.61$ result in the same SOM trajectories with an arc angle of ${90}^{\circ }$, a convenient reference point for comparing the geometric phase accumulated in each case. Here $z_{\ell }=z_{\text{rel}}{d}_{\text{nat}}$, with $d_{\text{nat}}$ given by Eq. (15b).

Figure 6

Beam cross sections before each lens. The information used to plot the trajectories on the sphere of modes (Figs. 3 and 4), motion along the fibers [Fig. 5], and trajectories through the supplementary product space [Figs. 5 and 5] can all be combined in Eq. (7) to produce beam cross sections that depend on the transit fraction $\zeta$. The beam magnitude has a cross section that does not depend on transit fraction, so a single magnitude cross section is plotted for each lens in (a) and (b). (c) and (d) Cross-sectional phase of the beam entering the lens. These look similar, but they are clearly not the same. This is because the lens positions are different, $z_{\text{rel},1}=1.41$ and $z_{\text{rel},2}=2.61$, the values used to produce Fig. 5.

Figure 7

Beam cross sections after each lens. The information used to plot the trajectories on the sphere of modes (Figs. 3 and 4), motion along the fibers [Fig. 5], and trajectories through the supplementary product space [Figs. 5 and 5] can all be combined in Eq. (7) to produce beam cross sections that depend on the transit fraction $\zeta$. (a) and (b) Corresponding cross-sectional phases for the beams exiting the lenses; these are clearly different. (c) and (d) Verification plots in which the thin-lens equation (26) is used to plot the phase cross sections directly. These plots are identical to the mode-based plots of (a) and (b).

Figure 8

Intersection of small-circle arc and geodesic. The geometric phase accumulated by beam transit through a lens is quantified using Eq. (35). This requires that the solid angle subtended by the moon-shaped sliver, shown here, between geodesic and small-circle arcs be evaluated. Equation (36) gives its solid angle.

Figure 9

Dynamic phase accumulation as a function of lens position. Equations (32) and (24) are combined to generate the dynamic phase accumulated by beam transits through lenses 1 and 2, followed by a projection operation, as plotted on the SOM in Fig. 3. This phase depends on lens position $z_{\ell }$. For lens positions of $z_{\ell 1}=0.59z_{Rp}$ and $z_{\ell 2}=1.77z_{Rp}$, the SOM trajectories are the same for each lens (${90}^{\circ }$ arc angles) and the dynamic phase is evaluated to be the same, $-31.8^{\circ }$, for the two lenses.

Figure 10

Total phase accumulation as a function of lens position. Equations (34) and (24) are combined to generate the total phase accumulated by beam transits through lenses 1 and 2, followed by a projection operation, as plotted on the SOM in Fig. 3. This phase depends on lens position $z_{\ell }$. For lens positions of $z_{\ell 1}=0.59z_{Rp}$ and $z_{\ell 2}=1.77z_{Rp}$, the SOM trajectories are the same for each lens (${90}^{\circ }$ arc angles) and the total phase is evaluated to be the same, $-35.3^{\circ }$, for the two lenses.

Figure 11

Geometric phase accumulation as a function of lens position. Equations (32) and (34) or, equivalently, (35) can be used to generate the geometric phase accumulated by beam transits through lenses 1 and 2, as plotted on the SOM in Fig. 3. This phase depends on lens position $z_{\ell }$. For lens positions of $z_{\ell 1}=0.59z_{Rp}$ and $z_{\ell 2}=1.77z_{Rp}$, the SOM trajectories are the same for each lens (${90}^{\circ }$ arc angles) and the geometric phase is evaluated to be the same for the two lenses.

Figure 12

Geometric phase as a function of arc subtended. The geometric phase accumulated by beam transit through a lens (35) can be used to plot the geometric phase accumulated as a function of the SOM arc subtended. The maximum magnitude for a single lens, $-26.4^{\circ }$, occurs for an arc angle of ${180}^{\circ }$, while a complete small circle would generate twice as much geometric phase, $-52.7^{\circ }$. This is equal to $-\frac{1}{2}$ the solid angle enclosed by a polar cap of polar angle $\pi /4$, as given by Eq. (37).

Figure 13

SOM trajectories for cylindrical lens transits. Trajectories generated by a lens with $\alpha =\pi /2$ are shown for several sets of initial positions (colors) and lens orientations $\eta$. The start of each trajectory is indicated with a small sphere of the same color: red, ${\underset{-}{r}}_{\mathrm{init}}=\{0,0,1\}$ and $\eta =\pi /4$; green, ${\underset{-}{r}}_{\mathrm{init}}=\{0,1,0\}$ and $\eta =\pi /2$; magenta, ${\underset{-}{r}}_{\mathrm{init}}=\{0.861,0.357,0.362\}$ and $\eta =0$; and blue, ${\underset{-}{r}}_{\mathrm{init}}=\{0.612,-0.353,0.707\}$, with $\eta =j\pi /8$ and $j\in [0,7]$. Dashing is a guide to the eye to more easily see that each trajectory lies on a circle.