Achromatic orbital angular momentum generator

Let us assume that a light beam undergoes total internal reflection at the surface of a medium having a relative refractive index of n (see figure 1(a)). The relative phase change between the two polarization components TE and TM is given by

$$\begin{eqnarray}&&\delta =2{\rm arctan} \left( \frac{{\rm cos} ({{\theta }_{i}})\sqrt{{\rm sin} {{({{\theta }_{i}})}^{2}}-{{n}^{2}}}}{{\rm sin} {{({{\theta }_{i}})}^{2}}} \right),\end{eqnarray} \tag{ 1 }$$

where ${{\theta }_{i}}$ is the angle of incidence [20]. The beam acquires a total relative phase change between these orthogonal polarizations for a sequence of total internal reflections inside the dielectric given by the following Jones matrix:

$$\begin{eqnarray}{{\mathcal{J}}_{0}}=\left( \begin{array}{ccccccccccccccc} 1 & 0 \\ 0 & {{e}^{i\delta }} \\ \end{array} \right).\end{eqnarray} \tag{ 2 }$$

In this expression, δ is the relative phase change between two polarization states after all reflections, which is a function of initial incidence angle, i.e. $\delta :=\delta ({{\theta }_{i}},n)$.

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The relative angle between the reflective surface and the incoming polarization defines the birefringence angle α, which is analogous to the orientation angle of the fast axis in a wave plate. Therefore, the polarization of the incoming beam after passing through the medium is modified by the orientation of the surface. The polarization dependent phase shift corresponding to a rotated surface can be calculated by means of the Jones calculus: ${{\mathcal{J}}_{\alpha }}=R(-\alpha )\cdot {{\mathcal{J}}_{0}}\cdot R(\alpha ),$ where $R(\alpha )$ is a 2 $\times$ 2 rotation matrix⁴. Reflection from a flat surface implies a uniform phase variation for a set of parallel rays; instead a curved surface introduces a nonuniform phase change. This nonuniform phase alteration comes from the nonuniformity of the birefringence angle, α, introduced by the curved surface. Thus, it gives rise to a nonuniform rotation of polarization, e.g. a linearly polarized beam is transformed into a specific polarization topology. The Jones operator of an infinitesimal reflective surface at angle α with respect to a fixed reference, say for example the horizontal axis, in the circular polarization basis, is given by:

$$\begin{eqnarray}{{\mathcal{J}}_{\alpha }}=\left( \begin{array}{ccccccccccccccc} {\rm cos} \left( \frac{\delta ({{\theta }_{i}},n)}{2} \right) & i{\rm sin} \left( \frac{\delta ({{\theta }_{i}},n)}{2} \right)\;{{e}^{-2i\alpha }} \\ i{\rm sin} \left( \frac{\delta ({{\theta }_{i}},n)}{2} \right)\;{{e}^{2i\alpha }} & {\rm cos} \left( \frac{\delta ({{\theta }_{i}},n)}{2} \right) \\ \end{array} \right).\end{eqnarray} \tag{ 3 }$$

In general, α depends on the transverse coordinates, e.g. $\alpha :=\alpha (r,\phi )$ in polar coordinates. However, in our case we assume that the angle of the reflective surface is only a function of the azimuthal angle, i.e. $\alpha (\phi )=q\;\phi$, where q is an integer or half-integer constant defined as the topological charge. A completely cylindrically symmetric reflective surface, such as that in figure 1(b), possesses a unit topological charge, i.e. $\alpha (\phi )=\phi$. This optic, consisting of two glued hollow axicons, generates light OAM of $\ell =\pm 2$, where the sign depends on the input polarization state. Let us assume a left-handed circularly polarized input beam, i.e. $|L\rangle ={{(1,0)}^{T}},$ where T stands for transpose operator. Then, the emerging beam can be split into two parts using equation (3), ${\rm cos} \left( \frac{1}{2}\;\delta ({{\theta }_{i}},n) \right)|L\rangle +i{\rm sin} \left( \frac{1}{2}\;\delta ({{\theta }_{i}},n) \right)\;{{e}^{2i\phi }}|R\rangle$, and in a similar manner it transforms a right-handed circularly polarized beam, i.e. $|R\rangle ={{(0,1)}^{T}}$, into ${\rm cos} \left( \frac{1}{2}\;\delta ({{\theta }_{i}},n) \right)|R\rangle +i{\rm sin} \left( \frac{1}{2}\;\delta ({{\theta }_{i}},n) \right)\;{{e}^{-2i\phi }}|L\rangle$. Thus the output polarization, as one would expect, depends on the relative optical retardation between the TE and TM polarizations. Importantly, when it is designed to behave as a half-wave plate, i.e. $\delta =\pi$, the device does transform a left-circularly polarized beam into a right-circularly polarized beam with an OAM value of $\ell$ = +2. Inversely, it transforms the right-circularly polarized beam into a left-circularly polarized beam, and gains an OAM value of $\ell =-2$. A circularly polarized light beam carries angular momentum of $\pm \hbar$ per photon. Since the device is cylindrically symmetric there should be no exchange of angular momenta with the device. In the first case, assuming a left-circularly polarized TEM₀₀ input beam, the total value of angular momentum is $+\hbar$ since SAM = $+\hbar$ and OAM = 0. The device inverts the polarization helicity into $-\hbar$, and therefore gains an OAM of $+\hbar -(-\hbar )\;=\;+2\hbar$, since total angular momentum must be conserved. Instead, for the case of a right-circularly polarized input beam, the sign changes to a negative. Indeed, the device works as a full spin-to-orbital angular momentum converter. The physics of this effect lies at the heart of a well-known quantum phenomena known as the geometric phase, where the polarization state of photons evolves adiabatically in the Hilbert space, and it acquires an additional phase that is given by geometrical features of the Hilbert space. For our case, this geometrical phase is proportional to half of the solid angle of the evolution path in the polarization Poincaré sphere [21]. This feature has already been well-investigated in a patterned dielectric, a patterned liquid crystal cell device, and very recently in a patterned plasmonic metasurface [22–24].

The achromatic spin-to-orbital angular momentum coupler with a specific optical retardation value of π can also be used to generate a vector vortex beam. These types of beams have specific nonuniform transverse polarization patterns. A subclass of these beams, cylindrical vector beams, corresponds to a superposition of two circularly polarized light beams with opposite OAM values, i.e. $|L\rangle {\rm exp} (-i\ell \phi )+{{{\rm e}}^{i\gamma }}|R\rangle {\rm exp} (+i\ell \phi )$, where γ is a relative phase between these two beams. In the case of $|\ell |=2$, these beams can be generated by feeding our device with a linearly polarized input beam. In the circular polarization basis, linear polarization states can be expressed as the following superposition states: $|\theta \rangle =({{e}^{-i\theta }}|L\rangle +{{e}^{i\theta }}|R\rangle )/\sqrt{2}$, where θ is the angle of linear polarization with respect to the horizontal axis. If we apply the Jones operator ${{\mathcal{J}}_{\alpha }}$ to a specific linear polarization state, e.g. horizontal polarization $|H\rangle \;:=\;|\theta =0\rangle ,$ we obtain the following output polarization state:

$$\begin{eqnarray}&&{{\mathcal{J}}_{\alpha }}\cdot |H\rangle ={\rm cos} (2\phi )|H\rangle +{\rm sin} (2\phi )|V\rangle ,\end{eqnarray} \tag{ 4 }$$

where a $\pi /2$ global phase is omitted. In this case we obtain a specific transverse polarization pattern as described in figure 2(b). It is possible to reconstruct this polarization pattern by performing polarization state tomography, where we measure the intensity of different polarization states using a sequence of a quarter-wave plate and a half-wave plate followed by a polarizing beam splitter. Furthermore, one can determine the OAM value from these types of measurements, as it can be shown that the number intensity petals (number of maxima) in the transverse profile is equal to twice the OAM value.

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The retardation introduced by the reflections, i.e. $\delta ({{\theta }_{i}},n)$, can be intentionally adjusted to not impose a full spin-to-orbital angular momentum conversion. Therefore, a fraction of the (circularly polarized) beam given by $\eta ={{\left| {\rm sin} \left( \frac{1}{2}\delta ({{\theta }_{i}},n) \right) \right|}^{2}}$ undergoes the full spin-to-orbital angular momentum conversion, where its polarization flips to opposite handedness and gains an OAM value of $|\ell |=2$. The remaining part, $1-\eta$, does not go through the spin-to-orbital angular momentum conversion process, and thus possesses the same polarization and OAM value as the input beam. Implementing specific retardations, in particular having a power balance between spin-to-orbital and non–spin-to-orbital angular momentum converted photons (i.e. $\eta =1/2$), leads to complex transverse polarization patterns called 'polarization-singular beams' or 'Poincaré beams' [25, 26]. This condition requires a static design of the device, for instance by changing the dielectric material or the cone angle, and cannot be adjusted dynamically.

Due to the presence of the surface singularity on the axis of the device (D₁ in figure 3(a)), a small portion of the beam near this region is scattered out. Thus, the transmitted beam possesses a doughnut shape at the exit face of the device. Nevertheless, since the emerging beam carries an OAM value of $\pm 2$, this doughnut shape remains unchanged by free-space propagation. The acceptance aperture of the truncated outer cone, D₂ in figure 3(a), introduces further diffraction on the beam, which affects the beamʼs radial shape similar to that presented in most OAM generators [27]. It is worth noticing that both effects are present in all OAM generator devices, except for the cylindrical mode convertor that transfers the cartesian laser cavity modes, i.e. Hermite–Gaussian, into the cylindrical solution of Laguerre–Gaussian modes [14]. Furthermore, the emerging beam undergoes a type of radial astigmatism, since the rays close to the center traverse faster than the rays at the periphery for an imperfectly collimated beam. However, this radial astigmatism can be compensated by adding a radially shaped pattern to the input and output ports of the device.

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Fabrication limitations encountered in this experiment consist of a central circular orifice and of an annular region where light is not transmitted, called the dead zone. The part of the beam that is transmitted through the central opening does not undergo the sequence of total internal reflections that lead to spin-to-orbital angular momentum coupling. This central non-converted region of the output beam can be filtered out with a sequence of a quarter-wave plate and a polarizing beam splitter, since the converted and non-converted beams have orthogonal polarizations. Only the outer annular region leads to spin-to-orbital angular momentum coupling, thus generating a beam carrying orbital angular momentum or a vector vortex beam.

Because we do not have access to the entire transverse profile of the output beam, the characteristic doughnut-shaped intensity profile is not visible. Thus, we determine the OAM value by using projective measurements on a generated vector beam with a horizontally polarized input beam. In the case of a spin-to-orbital angular momentum coupling device with a unit topological charge, we would expect to see a cross pattern after projecting the outgoing beam onto a linear polarizer. In general, the number of bright regions in the transverse intensity profile of the projective measurement, i.e. petals, is twice that of the OAM value of the beam. Rotating the polarizer causes a rotation onto the four-fold petal pattern. This set of polarization projective measurements constitutes an accurate means of determining the OAM value of an output beam in the case of a circularly polarized input beam. Furthermore, the reduced Stokes parameters, which are necessary to reconstruct the polarization pattern of the vector vortex beam, can be calculated from this set of measurements.

To show the broadband performance of the device, we perform polarization projective measurements for three different laser sources; namely red, green and blue (RGB). The red, green and blue laser sources correspond to a He–Ne laser at a wavelength of 633 nm and diode lasers operating at wavelengths of 532 nm and 405 nm, respectively. A polarizing beam splitter is used to obtain a horizontally polarized input beam. Then, horizontal, vertical, diagonal and anti-diagonal components of the intensity profile are measured using a sequence of a half-wave plate and a polarizing beam splitter. Figure 4 shows the desired 'cross' patterns previously described. The intensity profiles for left and right circularly polarized input beams are also shown in figure 4, where the intensity forms a uniform doughnut shape. As a measure of the quality of coupling between SAM and OAM in our device, polarization conversion efficiency (the amount of light converting from one polarization handedness to the opposite one in circular polarization basis) is measured. Conversion efficiencies of 96%, 95% and 94% for red, green and blue light are obtained, respectively. These conversion efficiencies are quite high and indicative of the high quality of the OAM generator.

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Polarization projective measurements shown in the previous section give a nice illustration of the spin-to-orbital angular momentum coupling nature of the device. In this section, we show quantitative measurements confirming the generation of OAM values of $2\hbar$ per photon for a left-handed circularly polarized beam, and $-2\hbar$ per photon for a right-handed circularly polarized beam. In order to reconstruct the OAM state of the output beam we performed OAM state tomography. We measured the OAM spectrum of the output beam by performing a phase-flattening projective measurement technique [29]. The measured OAM power spectrum is negligible for $|\ell |\ne 2$. The results confirm that the emerging beam from the device carries OAM values of $\ell =\pm 2$. Thus, we limit our analysis to a bi-dimensional OAM Hilbert space spanned by the following basis: ${{\{|\ell \;=\;+2\rangle }_{o}},|\ell =-2{{\rangle }_{o}}\}$, where $|\ell {{\rangle }_{o}}$ represents an OAM state in this Hilbert space. Hence the OAM state of the output beam can be represented by $|\psi {{\rangle }_{o}}=\alpha |+2{{\rangle }_{o}}+\beta |-2{{\rangle }_{o}}$, where α and β are complex numbers. This specific OAM sub-space is isomorphous to the polarization Hilbert space spanned by two orthogonal polarization states, which can be represented by the SU(2) group [30]. Pauli Matrices ${{\hat{\sigma }}_{x}}$, ${{\hat{\sigma }}_{y}}$, ${{\hat{\sigma }}_{z}}$, and the identity matrix $\hat{I}$ are corresponding generators of the SU(2) group. Therefore, one can reconstruct the OAM density matrix of the output light beam, i.e. ${{\rho }_{|\psi {{\rangle }_{o}}}}\;:=\;|\psi {{\rangle }_{o}}\langle \psi {{|}_{o}}$, by projecting the unknown OAM state $|\psi {{\rangle }_{o}}$ over the eigenstates of the generators. This is identical to measuring the polarization Stokes parameters, i.e. projecting the state on the H, V, A, D, L and R basis. The correspondence with the case of polarization is straightforwardly done by associating $|+2{{\rangle }_{o}}$ and $|-2{{\rangle }_{o}}$ to the left and right-handed circular polarized states of light, respectively. A full characterization of the unknown OAM state can be done by measuring four independent Stokes-like parameters ${{\Lambda }_{i}}$ ($i=0,1,2,3$) defined as:

$$\begin{eqnarray}&&\left\{ \begin{array}{ccccccccccccccc} {{\Lambda }_{0}}={{P}_{|+2{{\rangle }_{o}}}}+{{P}_{|-2{{\rangle }_{o}}}} \\ {{\Lambda }_{1}}={{P}_{|h{{\rangle }_{o}}}}-{{P}_{|v{{\rangle }_{o}}}} \\ {{\Lambda }_{2}}={{P}_{|a{{\rangle }_{o}}}}-{{P}_{|d{{\rangle }_{o}}}} \\ {{\Lambda }_{3}}={{P}_{|+2{{\rangle }_{o}}}}-{{P}_{|-2{{\rangle }_{o}}}}. \\ \end{array} \right.\end{eqnarray} \tag{ 5 }$$

Where ${{P}_{|i{{\rangle }_{o}}}}$ are the power of the projective measurement over the state of $|i{{\rangle }_{o}}$, and $|h{{\rangle }_{o}}=\left( |+2{{\rangle }_{o}}+|-2{{\rangle }_{o}} \right)/\sqrt{2}$, $|v{{\rangle }_{o}}=-i\left( |+2{{\rangle }_{o}}-|-2{{\rangle }_{o}} \right)/\sqrt{2}$, $|a{{\rangle }_{o}}=\left( |+2{{\rangle }_{o}}+i|-2{{\rangle }_{o}} \right)/\sqrt{2}$, and $|d{{\rangle }_{o}}=\left( |+2{{\rangle }_{o}}-i|-2{{\rangle }_{o}} \right)/\sqrt{2}$ are the eigenstates of the ${{\hat{\sigma }}_{x}}$ and ${{\hat{\sigma }}_{y}}$, respectively [31]. In order to achieve these sets of measurements, we implement a well-known phase-flattening projective measurement. In this technique, the emerging beam from the device is imaged onto a spatial light modulator (SLM) where different computer-generated holograms corresponding to the above six measurements were displayed. These holograms 'flatten' the corresponding complementary phase-front of the beam at the first order of diffraction. Thus, only the 'flattened' component of the optical field can be coupled into a single mode optical fibre, which results in measuring ${{P}_{|i{{\rangle }_{o}}}}$ [32, 33]. The SLM and the single mode optical fibre act as the polarizer in the case of polarization state tomography. We implement this approach using a Pluto HOLOEYE SLM suitable for visible, excluding ultra-violet, wavelengths. The power of the projective measurements ${{P}_{|i{{\rangle }_{o}}}}$ are recovered by a Newport power meter. From these measurements the Stokes-like parameters of equation (5) are calculated. The reconstructed OAM density matrix of the output beam is then calculated by the following expression:

$$\begin{eqnarray}&&\hat{\rho }=\frac{1}{2}\mathop{\sum }\limits_{i=0}^{3}\frac{{{\Lambda }_{i}}}{{{\Lambda }_{0}}}{{\hat{\sigma }}_{i}}.\end{eqnarray} \tag{ 6 }$$

Figure 5 shows the reconstructed OAM density matrix for different input polarization states, and a red and green laser source. Fidelity of 94% and 91% are observed for generating the OAM value of +2 and −2 with a red laser source, and 93% and 98% are observed for generating OAM value of +2 and −2 with a green laser source. It is worth mentioning that we do not have access to a UV-SLM since all commercial liquid-crystal devices will not work as desired for the case of blue/UV light. This is based on the fact that they are originally made using UV light, and consequently, illuminating them with a UV light source would change the orientation of the polymer substrate. However, the analysis reported in the previous section confirmed the existence of spin-to-orbital angular momentum coupling for the blue laser as well. Thus, our device can be used to generate OAM values of $\pm 2$ at the UV wavelength regime, which is a missing fact in the community.

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We examine the broadband performance of the device by performing a polarization projective measurement for the case of a linearly polarized white-light input beam. A liquid crystal display (laptop screen) is used as the polarized white light source and a high definition camera was used to detect the transverse intensity profile of the output beam after projecting the light polarization state by means of a broadband half-wave plate and a polarizing beam splitter. The 'cross' pattern was also observed for the white light source; rotating the half-wave plate leads to a rotation of the cross through the same angle.

Figure 6(a) shows the transverse intensity distribution of the outgoing beam from the device when illuminated with a white polarized source. As can be seen, the transmitted beam has a uniform intensity distribution. The petal pattern is also visible when the device is sandwiched between two crossed polarizers; see figure 6(b). Polychromatic effects around the null intensity lines are observed, which lies at the heart of diffraction. When the device is illuminated with a polychromatic (white) wave, different colours undergo different diffraction effects since they have different wavelengths [34, 35]. As we have already discussed, the device transfers a linearly polarized beam into a vector vortex beam. We reconstructed the polarization pattern of the generated vector beam by performing polarization state tomography, where the reduced Stokes parameters of the output beam are measured via polarization projective measurements; see [28] for more details. The reconstructed topology of the generated beam is shown in figure 6(c). As can be seen, the transverse polarization pattern corresponds to the expected 'dipole' shape shown previously in the theory section.

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