Abstract
Wave retarders having spatially varying optical axes orientations, called qplates are extremely efficient devices for converting spin to orbital angular momentum of light and for the generation of optical vortices. Most often, these plates are designed for a specific wavelength and have a homogeneous constant retardance. The present work provides a polarimetric approach for overcoming both these limitations. We theoretically propose and experimentally demonstrate qplates with tunable retardance, employing a combination of only standard qplates and waveplates. A clear prescription is provided for realizing wavelength indepedent qplates for a desired retardance, with a potential for ultrafast switching. Apart from the potential commercial value of the proposed devices, our results may find applications in quantum communication protocols, astronomical coronography, angular momentum sorting and in schemes that leverage optical vortices and spin to orbital angular momentum conversion.
Introduction
Light beams having a heterogeneous distribution of polarization in their transverse plane are called vector beams^{1}. Such vector beams have widespread applications in areas ranging from optical tweezing^{2}, in achieving sharper focusing^{3} and for the generation of higherdimensional quantum states of photons for use in quantum key distribution protocols^{4,5}. In the last couple of decades, vector beams with a helical wavefront, possessing an optical singularity in their beam axis, called vectorvortex beams or “twisted light”, have been extensively investigated^{6}. These beams carry an orbital angular momentum (OAM)^{7} which is to be distinguished from spin angular momentum (SAM) associated with circularly polarized light^{8}. Helical beams are generated in a variety of ways: using spiral wave plate^{9}, subwavelength gratings^{10}, cylindrical lenses^{11}, holographic patterns^{12,13}, spatial light modulators^{14} and plasmonic metasurfaces^{15}.
Another efficient way is through the use of specially engineered liquid crystal based waveplates called qplates, which have a uniform retardance but with spatially varying optic axis^{16}. The advent of qplates has opened the flood gates of research and applications that exploit the inter play of SAM and OAM^{17}. Conventionally, the optic axis orientation, α, of qplates varies linearly with the azimuthal angle, \(\varphi \), given by \(\alpha (\varphi )=q\varphi +{\alpha }_{0}\). Such qplates are defined by three parameters \((q,{\rm{\Gamma }},{\alpha }_{0})\) where, q is an integer or a halfinteger, called the topological charge, \({\rm{\Gamma }}\) represents the retardance, and \({\alpha }_{0}\) the offset angle. Although standard qplates are predominantly of halfwave retardance, those with a retardance \({\rm{\Gamma }}\ne \pi \) have also found important applications. For instance, using a \({\rm{\Gamma }}=\frac{\pi }{2}\) qplate, full Poincare beams^{18} have been realized^{19}. On the other hand, a circularly polarized light through a qplate having \({\rm{\Gamma }}=\pi \) (halfwave qplate hereafter) converts into circularly polarized light of opposite helicity, in addition to picking up an OAM of magnitude \(2q\hslash \) per photon. This process is referred to as SAM to OAM conversion (STOC), and has found numerous applications^{17,20,21}. For many of these applications, it is important to have a precise control over the fraction of light undergoing STOC. For instance, controlled STOC has been employed for realizing quantum random walks, where the SAM functions as the coin space, while the OAM functions as the walk space^{22,23,24,25}. In these experiments, the fraction of light undergoing STOC was controlled by tuning the retardance of the qplates using an externally applied voltage^{26}. Retardance of the qplates can also be tuned by varying the temperature^{27}, but this method suffers from a very slow response time. One could also use the intensity of the light itself to control the STOC, but this is a nonlinear phenomenon^{28,29,30}.
Retardance of a qplate depends strongly on the wavelength of the incident light, while its topological charge and offset angle remain independent of it. Quite often, commercially available standard qplates exhibit the advertised retardance only at around the design wavelength. Indeed, the performance of a commercial qplate at wavelengths different from the operational wavelength has been recently studied in^{31}. Of late, generation of achromatic optical vortices has been recieving wide interest, owing to their applications primarily in the field of astronomical coronagraphy^{32,33,34,35}. While different achromatic methods of achieving optical vortices have been proposed^{36,37,38,39}, achromatic qplates have received scant attention^{40,41,42,43,44,45}.
Achromaticity and retardance tunability of homogeneous waveplates have been achieved using a sequence of waveplates of appropriate retardances and orientations^{46,47,48,49}. Motivated by this idea, we seek to design “effective qplates”, whose retardance is tunable across a broad range of wavelengths, using a combination of qplates and waveplates. By effective qplate we mean, the sequence of qplates is describable as a single qplate. In other words: (i) The resultant optic axis of the sequence \({\alpha }_{e}(\varphi )\) also varies linearly with the azimuthal angle, \({\alpha }_{e}(\varphi )={q}_{e}\varphi +{\alpha }_{0e}\) and (ii) The resultant retardance of the sequence \({{\rm{\Gamma }}}_{e}\) is independent of \(\varphi \).
In this paper, we study the effective qplates emerging from a sequence of three qplates. We propose two distinct nontrivial means of realizing these effective qplates. In each of these cases, we prove that the retardance of the effective qplates is tunable. We experimentally validate this result by demonstrating fractional STOC, which hitherto has been possible only by varying applied voltage or temperature. In addition, we show that the retardance tunability of these effective qplates can be extended to a broad range of wavelengths.
Results
For convenience, we represent a qplate with parameters \((q,{\rm{\Gamma }},{\alpha }_{0})\) by the notation \({W}_{{\rm{\Gamma }}}(q,{\alpha }_{0})\). This notation offers the flexibility of representing even the homogeneous waveplates (like halfwave and quarterwave plates), where the orientation of the optic axis remains constant. A homogeneous waveplate with retardance \({\rm{\Gamma }}\) and optic axis orientation \(\alpha \) is represented in this notation as \({W}_{{\rm{\Gamma }}}(0,\alpha )\).
Consider three qplates \({W}_{{{\rm{\Gamma }}}_{1}}({q}_{1},{\alpha }_{01})\), \({W}_{{{\rm{\Gamma }}}_{2}}({q}_{2},{\alpha }_{02})\) and \({W}_{{{\rm{\Gamma }}}_{3}}({q}_{3},{\alpha }_{03})\) arranged in a sequence. It can be seen that if
then condition (i) of realizing effective qplates is satisfied, independent of parameters of the central qplate. The effective retardance \({{\rm{\Gamma }}}_{e}\), when constraints of eq. (1) are satisfied, is
where \({\rm{\Delta }}q={q}_{2}{q}_{1}\) and \({\rm{\Delta }}{\alpha }_{0}={\alpha }_{02}{\alpha }_{01}\) (see Methods for details). Condition (ii), that the effective retardance \({{\rm{\Gamma }}}_{e}\) is independent of azimuthal angle \(\varphi \), is satisfied under the following two nontrivial cases:

Case (a): The outer plates have a retardance \({{\rm{\Gamma }}}_{1}\) equal to oddmultiple of π.

Case (b): The central and outer qplates have identical topological charge, \({\rm{\Delta }}q=0\).
These two cases are depicted in Fig. 1.
Case (a): Effective qplates realized using halfwave qplates
Consider first the arrangement as shown in Fig. 1a. A qplate \({W}_{{{\rm{\Gamma }}}_{2}}({q}_{2},{\alpha }_{02})\) is placed in between two identically oriented halfwave qplates \({W}_{\pi }({q}_{1},{\alpha }_{01})\). The effective retardance \({{\rm{\Gamma }}}_{e}\), the effective topological charge \({q}_{e}\) and effective offset angle \({\alpha }_{0e}\) in this case are given by (see Methods for details):
In short,
The effective topological charge is \({q}_{e}=2{q}_{1}{q}_{2}\). This arrangement can therefore be used for synthesizing qplates of different topological charges. For instance, using commercially available qplates of topological charge 1 and 0.5, it is possible to realize an effective qplate with topological charge \({q}_{e}=1.5\). Similarly, placing \({W}_{{\rm{\Gamma }}}({q}_{2},{\alpha }_{0})\) between two homogeneous halfwave plates \({W}_{\pi }(0,0)\) yields an effective qplate \({W}_{{\rm{\Gamma }}}({q}_{2},\,{\alpha }_{0})\), where the topological charge is inverted. These results, but restricted only to halfwave qplates have been reported in^{50,51,52}. The current analysis generalizes them to arbitrary retardance qplates.
The offset angle of the qplates, in general, can be altered by merely rotating them. The only exception to this is qplates with \(q=1\), owing to their rotational symmetry of optic axis^{16}. Hitherto, experiments involving \({W}_{{\rm{\Gamma }}}(1,{\alpha }_{0})\) qplates necessitated fabricating distinct qplates for each \({\alpha }_{0}\)^{53}. This requirement can be mitigated by realizing an effective \({W}_{{\rm{\Gamma }}}(1,{\alpha }_{0})\) plate. One way towards this is through \({W}_{\pi }(0.5,0)\to {W}_{{\rm{\Gamma }}}(0,\,\alpha )\to {W}_{\pi }(0.5,0)\). This yields an effective qplate \({W}_{{\rm{\Gamma }}}(1,\alpha )\) (see eq. (4)), the offset angle of which can be changed by merely rotating the central homogeneous waveplate \({W}_{{\rm{\Gamma }}}(0,\,\alpha )\).
From eq. (4), the retardance of the effective qplate is equal to the retardance of the central qplate. This result holds good even if the central plate is homogeneous (\({q}_{2}=0\)). It is therefore possible to realize a qplate of any retardance, by sandwiching a homogeneous waveplate of that retardance between two halfwave qplates. Further, this result can be utilized for realizing a tunable retardance qplate \({{\rm{\Gamma }}}_{e}\in (0,2\pi )\), provided retardance of the central homogeneous waveplate is itself tunable, \({{\rm{\Gamma }}}_{2}\in (0,2\pi )\). Retardance tunable homogeneous waveplates have been realized in many ways^{46,47,54,55,56}. By replacing the central plate of Fig. 1a with any such tunable retarder yields an effective qplate with tunable retardance.
As an illustration, here we demonstrate tunable retardance qplate, by sandwiching a Berek plate^{55} between two \(q=1\) halfwave qplates. Berek plate was set for five different retardances, \({\rm{\Gamma }}\in (0,\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4},\pi )\) while its orientation \({\alpha }_{02}\) was fixed at 0. The performance of this arrangement is validated against that of the \({W}_{{\rm{\Gamma }}}(2,0)\) qplate, for plane and circularly polarized light beams.
At every retardance of the Berek plate, the vertically polarized light beam is sent through the aforementioned arrangement and the intensity profile in the transverse plane of the evolved vector beam is recorded. Further, to validate the polarization distribution, the evolved beam was projected into six cardinal directions, horizontal (H), vertical(V), diagonal (D), antidiagonal (A), left circular (L) and right circular (R). The theoretical intensity profiles obtained when a vertically polarized light beam exits a \({W}_{{\rm{\Gamma }}}(2,0)\) qplate, set for the same retardances, and evolves for the same distance as in the experiment, is computed using the formulae derived in references^{19,57}. The experimental and the corresponding theoretical images are shown in Fig. 2.
We observe eight lobes in the plane polarization projections intensity profiles, indicating an effective topological charge \({q}_{e}=2\), which is double that of the individual qplates, as expected from eq. (4). These lobes are identical and of equal intensity for \({{\rm{\Gamma }}}_{e}=\pi \). Further, the unprojected intensity at the beam center gradually falls with increasing retardance \({{\rm{\Gamma }}}_{e}\), completely vanishing when the retardance is \({{\rm{\Gamma }}}_{e}=\pi \). Further, since the input beam is vertically polarized, the intensity profile in its orthogonal projection, that is along the horizontal component is always a doughnut with singularity at the center, for any retardance. At \({{\rm{\Gamma }}}_{e}=\pi \), the horizontal and vertical projections are identical and have equal power. The excellent match between the intensity profile obtained with our gadget and the theoretically expected profile confirms the suitability of our setup for obtaining qplates with tunable retardance.
A major application of tunable retardance qplates has been in obtaining a controlled spin to orbital conversion of angular momentum of light. Light beam having a definite spin (polarization) and orbital angular momentum sent through a qplate of retardance \({\rm{\Gamma }}\ne \pi \), gets converted into a superposed beam having no definite spin or orbital angular momentum. Denoting the left (right) circularly polarized beam with OAM of \(m\hslash \) units by \(L,m\rangle \) \((R,m\rangle )\), the action of a qplate \({W}_{{\rm{\Gamma }}}(q,{\alpha }_{0})\) is given by:
The STOC fraction is obtained by measuring the power in the left and right circular projections. Fraction of the beam that remains in the initial state \(L,0\rangle \) is proportional to \({\cos }^{2}(\frac{{\rm{\Gamma }}}{2})\), while the fraction getting converted to the \(R,2\rangle \) state is proportional to \({\sin }^{2}(\frac{{\rm{\Gamma }}}{2})\). Complete spin to angular momentum conversion is possible only when \({\rm{\Gamma }}=\pi \).
Here we demonstrate STOC using the tunable \({W}_{{\rm{\Gamma }}}(2,0)\) qplate discussed above, using left circularly polarized Gaussian beam \(L,0\rangle \). Figure 3 shows the power fraction in the left and right circular projections, as a function of the retardance of the central Berek plate (and hence the effective retardance of the qplate).
In Fig. 3, stars and diamonds indicate the experimentally measured average power fractions in left and right circular projections respectively, at different retardances of the central Berek plate. At every retardance, averaging is done over 30 readings, with each reading lasting for 1 second. The average power fractions are plotted together with their errorbars. Errorbars are the maximum and minimum values of these 30 readings, which happen to be less than 0.1% at every data point, hence too small (smaller than the symbols) to be visible. The continuous lines are the theoretical expectations obtained from eq. (5). Power fraction in left and right circular projections show sinusoidal variation with the retardance of Berek plate, becoming equal at odd multiples of \(\frac{\pi }{2}\). The close agreement between theory and experiments demonstrates the excellent functionality of this gadget as a qplate with tunable retardance.
Case (b): Effective qplate realized by using three qplates of identical topological charge
Here, we analyze case (b) depicted in Fig. 1b, where the condition for obtaining the effective qplate with uniform retardance is \({\rm{\Delta }}q=0\): all three qplates have identical topological charge. From eq. (2), the effective retardance \({{\rm{\Gamma }}}_{e}\) in this case is decided by \({\rm{\Delta }}{\alpha }_{0}\), being \(2{{\rm{\Gamma }}}_{1}+{{\rm{\Gamma }}}_{2}\) for \({\rm{\Delta }}{\alpha }_{0}=0\) and \(2{{\rm{\Gamma }}}_{1}{{\rm{\Gamma }}}_{2}\) for \({\rm{\Delta }}{\alpha }_{0}=\,\frac{\pi }{2}\,\). The full span of \((0,2\pi )\) retardance is possible using this arrangement when \(2{{\rm{\Gamma }}}_{1}{{\rm{\Gamma }}}_{2}=0\) and \(2{{\rm{\Gamma }}}_{1}+{{\rm{\Gamma }}}_{2}=2\pi \), that is when \({{\rm{\Gamma }}}_{1}=\frac{\pi }{2}\) and \({{\rm{\Gamma }}}_{2}=\pi \). The effective retardance \({{\rm{\Gamma }}}_{e}\) and effective orientation \({\alpha }_{0e}\) in this case becomes:
In our notation,
Recall that \({\rm{\Delta }}{\alpha }_{0}={\alpha }_{02}{\alpha }_{01}\) is relative orientation of the optical axes of outer and central plates at the zero azimuth. Since for qplates with \(q\ne 1\), it is possible to change the offset angle \({\alpha }_{0}\) by changing their orientation, the effective retardance can be tuned just by rotating either the central or the outer qplates. It is hence possible to construct a qplate of tunable effective retardance \({\rm{\Gamma }}\in (0,2\pi )\) using two quarterwave qplates and a halfwave qplate, all having identical topological charge (\(\ne 1\)). This is yet another way of realizing qplate with tunable retardance involving three qplates.
For qplates with different retardances, \({{\rm{\Gamma }}}_{1}(\ne \frac{\pi }{2})\) and \({{\rm{\Gamma }}}_{2}(\,\ne \,\pi )\), it may not be possible to achieve the complete span of effective retardance \({{\rm{\Gamma }}}_{e}\in (0,2\pi )\). We briefly explore this scenario further in this section. It follows from eq. (2) that a retardance \({{\rm{\Gamma }}}_{e}\) is realizable, by a suitable \({\rm{\Delta }}{\alpha }_{0}\), provided the following condition is satisfied:
Figure 4 shows the possible values of \({{\rm{\Gamma }}}_{1}\) and \({{\rm{\Gamma }}}_{2}\) using which an effective retardance of (a) \({{\rm{\Gamma }}}_{e}=\frac{\pi }{2}\) and (b) \({{\rm{\Gamma }}}_{e}=\pi \) can be realized. The color coding is the value of \({\rm{\Delta }}{\alpha }_{0}\) for realizing them. For instance, it is possible to realize \({{\rm{\Gamma }}}_{e}=\frac{\pi }{2}\) with outer qplates having \({{\rm{\Gamma }}}_{1}=\frac{\pi }{4}\) and the central qplate having \({{\rm{\Gamma }}}_{2}=\frac{3\pi }{4}\), while it is not possible to achieve this effective retardance with \({{\rm{\Gamma }}}_{1}=\frac{3\pi }{4}\) and \({{\rm{\Gamma }}}_{2}=\frac{\pi }{4}\).
It would be interesting to know whether it is possible to obtain a desired retardance \({{\rm{\Gamma }}}_{e}\) using three qplates with identical retardances. The dotted line in the figure is \({{\rm{\Gamma }}}_{1}={{\rm{\Gamma }}}_{2}\) and its intersection with the colored region indicates whether the effective retardance \({{\rm{\Gamma }}}_{e}\) can be realized with three waveplates of identical retardance. For instance, since the point \((\frac{\pi }{4},\frac{\pi }{4})\) lies outside of the colored region in Fig. 4b, it is not possible to realize \({{\rm{\Gamma }}}_{e}=\pi \) using three qplates with \({{\rm{\Gamma }}}_{1}={{\rm{\Gamma }}}_{2}=\frac{\pi }{4}\). On the other hand, \((\frac{\pi }{2},\frac{\pi }{2})\) lies within the colored region, and hence it is possible to achieve \({{\rm{\Gamma }}}_{e}=\pi \) with \({{\rm{\Gamma }}}_{1}={{\rm{\Gamma }}}_{2}=\frac{\pi }{2}\). Figure 5 shows the span of effective retardance \({{\rm{\Gamma }}}_{e}\) achievable using three identical qplates having retardance \({\rm{\Gamma }}\). This is computed from eq. (2), by setting \({{\rm{\Gamma }}}_{1}={{\rm{\Gamma }}}_{2}={\rm{\Gamma }}\), \({\rm{\Delta }}q=0\) and by varying \({\rm{\Delta }}{\alpha }_{0}\) from 0 to \(\frac{\pi }{2}\). From the plot, it is evident that for \({\rm{\Gamma }} < \pi \), the minimum \({{\rm{\Gamma }}}_{e}\) is \({\rm{\Gamma }}\), while for \({\rm{\Gamma }} > \pi \) the maximum possible \({{\rm{\Gamma }}}_{e}\) is \({\rm{\Gamma }}\). With \({\rm{\Gamma }}=\pi \), the only possible effective retardance is \({{\rm{\Gamma }}}_{e}=\pi \).
While in case(a) the retardance tunability was achieved by replacing the central qplate by a tunable homogeneous waveplate, in case(b) the retardance tunability is achieved by merely rotating the central qplate. The possible span of realizable effective retardance in the latter case depends upon the retardances of the outer and the central plates. In the extreme case, the effective qplate realized using three halfwave qplates is just another halfwave qplate, offering zero tunability. Since standard commercial qplates are often of halfwave retardance, this arrangement may appear to be of little practical use towards realizing tunable retardance qplates. However, this sequence of three halfwave qplates will be useful for realizing wavelengthadaptable qplates, by which we mean an effective qplate which can be tuned to exhibit any desired retardance at any desired wavelength.
Realization of wavelengthadaptable qplates
We now examine the possibility of realizing wavelengthadaptability of qplates in two cases, (a) and (b). In case (a), for a wavelength different from the operating wavelength of the qplates, the outer two qplates have a retardance \({{\rm{\Gamma }}}_{1}\) different from \(\pi \), and hence from eq. (2), the effective retardance becomes a function of the azimuthal angle. So the threeplate arrangement of case (a) fails to act like an effective qplate at wavelengths different from the operating one.
In case (b), on the other hand, the condition for realizing an effective qplate is \({\rm{\Delta }}q=0\), which is a wavelengthindependent constraint. An effective qplate \({W}_{{{\rm{\Gamma }}}_{e}}({q}_{e},{\alpha }_{0e})\) at one wavelength continues to remain an effective qplate at a different wavelength, albeit with a different \({{\rm{\Gamma }}}_{e}\) and \({\alpha }_{0e}\). So we explore the possibility of wavelengthindependence in this case. For simplicity, consider three identical qplates having halfwave retardance at a design wavelength \({\lambda }_{d}\): \({\rm{\Gamma }}({\lambda }_{d})=\pi \). At a wavelength \(\lambda \) different from \({\lambda }_{d}\), the retardance \({\rm{\Gamma }}(\lambda )\) of each plate is different from \(\pi \). From Fig. 5, it is seen that an effective retardance of \({{\rm{\Gamma }}}_{e}=\pi \) is achievable for any retardance \({\rm{\Gamma }}\) within the window \([\frac{\pi }{3},\frac{5\pi }{3}]\). Hence, at any wavelength \(\lambda \) such that \(\frac{\pi }{3} < {\rm{\Gamma }}(\lambda ) < \frac{5\pi }{3}\), it is possible to realize an effective retardance of \(\pi \), using the arrangement of Fig. 1b. Similarly, an effective retardance of \({{\rm{\Gamma }}}_{e}=\frac{\pi }{2}\) is achievable for \({\rm{\Gamma }}\) within \([\frac{\pi }{6},\frac{\pi }{2}]\) or \([\frac{7\pi }{6},\frac{3\pi }{2}]\). So at any wavelengths \(\lambda \) such that \(\frac{\pi }{6} < {\rm{\Gamma }}(\lambda ) < \frac{\pi }{2}\) or \(\frac{7\pi }{6} < {\rm{\Gamma }}(\lambda ) < \frac{3\pi }{2}\), it is possible to realize a retardance of \(\frac{\pi }{2}\) using the arrangement of Fig. 1b.
For concreteness of this idea, we discuss here the possibility of realizing an effective qplate having retardance of \({{\rm{\Gamma }}}_{e}=\pi \) over a range of wavelengths, using three standard commercial qplates (Thorlabs, WPV10633, with q = 1) designed to provide a retardance \({\rm{\Gamma }}=\pi \) at 633 nm. The \({\rm{\Gamma }}(\lambda )\) plot for these qplates has been reported recently^{31}. While this study was carried out for \(q=1\) qplates, we proceed with the reasonable assumption that the similar dependence holds true even for \(q\ne 1\) qplates. There it was observed that \({\rm{\Gamma }}(\lambda )\) varies inversely with λ, with \({\rm{\Gamma }}(450\,{\rm{nm}})=\frac{3\pi }{2}\) and \({\rm{\Gamma }}(1050\,{\rm{nm}})=\frac{\pi }{2}\). Since the retardance variation of this qplate is confined between \(\frac{\pi }{2}\) and \(\frac{3\pi }{2}\), which is within the \([\frac{\pi }{3},\frac{5\pi }{3}]\) window, an effective retardance of \({{\rm{\Gamma }}}_{e}={\rm{\pi }}\) can be realized at any wavelength within the range \([450\,{\rm{nm}},1050\,{\rm{nm}}]\). Figure 6 shows the value of \({\alpha }_{02}\), at which the central qplate needs to be oriented, at every wavelength λ to obtain an effective retardance of π.
Figure 7 consolidates the possible retardances realizable at different wavelengths, using three Thorlabs WPV10633 qplates. The colorcoding is for the value of \({\rm{\Delta }}{\alpha }_{0}\). It is evident that a full wavelengthindependence is possible only for halfwave retardance \({{\rm{\Gamma }}}_{e}=\pi \). For other retardances, only piecewise wavelengthindependence is achievable.
The Fig. 7 can also be used for knowing the possible values of retardances realizable at every wavelength using three of the above qplates. For instance, while the retardance of π is realizable at every wavelength, close to the design wavelength it is only possible to realize an effective retardance of π.
Discussion
To summarize, an exhaustive study of the conditions under which a combination of three qplates function as an effective qplate has been carried out. The central result of this work is the identification of two inequivalent configurations for obtaining effective qplates. First configuration restricts the outer qplates to be of halfwave retardance, while the second configuration requires the three qplates to be of identical topological charge. We have shown that it is possible to tune the retardance of the effective qplate realized by either means, former method being less restrictive than the latter.
As an experimental demonstration towards this, we have realized a retardance tunable qplate by sandwiching a Berek plate between two halfwave qplates. The generated vectorvortex beams and the fractional STOC effect are found to compare excellently with theory, validating the practical utility of our method in realizing tunable retardance qplates. As against the current state of the art, the retardance tunability is achieved here through passive means. This retardance tunability can also be realized using only offtheshelf qplates and waveplates, by replacing the Berek plates with a gadget involving a halfwave plate sandwiched between two identically oriented quarterwave plates. On the other hand, by replacing the Berek plate with a fast acting Pockels cell, one could easily achieve submicrosecond switching times whereas conventional voltage or temperature tunable qplates have switching time in the order of milliseconds.
The effective qplates realized in the second configuration, while utilizing only standard wavelength sensitive qplates, are shown to function over a wide gamut of wavelengths.
While we have restricted our attention to standard qplates, our results are general and are applicable even to the so called metaqplates^{58}. We believe that the retardance tunability, control of topological charge and wavelengthindependence offered by our method will be of substantial commercial value towards realizing quantum protocols like quantum key distribution, quantum random walks and will lead to further exploration of optical phenomena involving interplay of SAM and OAM degrees of freedom.
Methods
The action of a waveplate with retardance \({\rm{\Gamma }}\) and optic axis oriented at an angle \(\alpha \) is described by the Jones Matrix \({U}_{{\rm{\Gamma }}}(\alpha )\)^{59}:
The diagonal elements of this matrix are complex conjugates, while the offdiagonal elements are purely imaginary and equal to each other. The resulting action due to a sequence of waveplates is obtained by the product of the Jones matrix of each of them. The offdiagonal elements of the resulting product matrix need not be purely imaginary. We term a sequence of waveplates as an “effective waveplate”, provided the offdiagonal elements of the product matrix are purely imaginary, as in eq. (8).
Consider three waveplates having retardance \({{\rm{\Gamma }}}_{j}\) and orientated at an angle \({\alpha }_{j},j=1,2,3\). The resultant matrices \({U}_{{{\rm{\Gamma }}}_{1}}({\alpha }_{1}){U}_{{{\rm{\Gamma }}}_{2}}({\alpha }_{2}){U}_{{{\rm{\Gamma }}}_{3}}({\alpha }_{3})\) represents an effective waveplate if \({{\rm{\Gamma }}}_{1}={{\rm{\Gamma }}}_{3}\) and \({\alpha }_{1}={\alpha }_{3}\)^{46}. The effective retardance \({{\rm{\Gamma }}}_{e}\) and effective orientation \({\alpha }_{e}\) of this effective waveplate are obtained by:
where σ_{x} is the Pauli spin matrix \([\begin{array}{ll}1 & 0\\ 0 & \,1\end{array}]\).
For a qplate, \({W}_{{\rm{\Gamma }}}(q,{\alpha }_{0})\), the Jones matrix is same as eq. (8), with \(\alpha =q\varphi +{\alpha }_{0}\). The condition \({\alpha }_{{}_{1}}={\alpha }_{3}\) translates to \({q}_{1}={q}_{3}\) and \({\alpha }_{01}={\alpha }_{03}\).
Experimental details
The experiments are carried out using qplates of Thorlabs make (model number: WPV10633, with q = 1). The Berek plate is of Newport make (model number: 5540). A HeNe laser of JDSU make (model number: 1145P) has been employed. Polarized light beam from this source is then expanded to a diameter of about 4 mm, from a combination of achromatic doublet lenses. For carrying out circular projections, achromatic quarterwave plates of Thorlabs make have been used. Linear polarizer used is of Thorlabs make, which has an extinction ratio of 100,000:1 for the light of 633 nm wavelength. The intensity profile in the transverse plane is captured at a distance of 1 m using a beam profiler of Newport make (model number: LBP2HRVIS). The optical power in the STOC experiments is measured using a photodiode sensor of Ophir make (model number PD300UV).
References
 1.
Qiwen, Z. Vectorial optical fields: Fundamentals and applications (World scientific, 2013).
 2.
Padgett, M. & Bowman, R. Tweezers with a twist. Nature photonics 5, 343 (2011).
 3.
Dorn, R., Quabis, S. & Leuchs, G. Sharper focus for a radially polarized light beam. Physical review letters 91, 233901 (2003).
 4.
Erhard, M., Fickler, R., Krenn, M. & Zeilinger, A. Twisted photons: new quantum perspectives in high dimensions. Light: Science & Applications 7, 17146 (2018).
 5.
Sit, A. et al. Highdimensional intracity quantum cryptography with structured photons. Optica 4, 1006–1010 (2017).
 6.
Zhan, Q. Cylindrical vector beams: from mathematical concepts to applications. Advances in Optics and Photonics 1, 1–57 (2009).
 7.
Allen, L., Beijersbergen, M. W., Spreeuw, R. & Woerdman, J. Orbital angular momentum of light and the transformation of laguerregaussian laser modes. Physical Review A 45, 8185 (1992).
 8.
Poynting, J. H. et al. The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light. Proc. R. Soc. Lond. A 82, 560–567 (1909).
 9.
Beijersbergen, M., Coerwinkel, R., Kristensen, M. & Woerdman, J. Helicalwavefront laser beams produced with a spiral phaseplate. Optics Communications 112, 321–327 (1994).
 10.
Biener, G., Niv, A., Kleiner, V. & Hasman, E. Formation of helical beams by use of pancharatnam–berry phase optical elements. Optics letters 27, 1875–1877 (2002).
 11.
Beijersbergen, M. W., Allen, L., Van der Veen, H. & Woerdman, J. Astigmatic laser mode converters and transfer of orbital angular momentum. Optics Communications 96, 123–132 (1993).
 12.
Heckenberg, N., McDuff, R., Smith, C. & White, A. Generation of optical phase singularities by computergenerated holograms. Optics letters 17, 221–223 (1992).
 13.
Mair, A., Vaziri, A., Weihs, G. & Zeilinger, A. Entanglement of the orbital angular momentum states of photons. Nature 412, 313 (2001).
 14.
Maurer, C., Jesacher, A., Fürhapter, S., Bernet, S. & RitschMarte, M. Tailoring of arbitrary optical vector beams. New Journal of Physics 9, 78 (2007).
 15.
Karimi, E. et al. Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface. Light: Science & Applications 3, e167 (2014).
 16.
Marrucci, L., Manzo, C. & Paparo, D. Optical spintoorbital angular momentum conversion in inhomogeneous anisotropic media. Physical review letters 96, 163905 (2006).
 17.
Marrucci, L. et al. Spintoorbital conversion of the angular momentum of light and its classical and quantum applications. Journal of Optics 13, 064001 (2011).
 18.
Beckley, A. M., Brown, T. G. & Alonso, M. A. Full poincaré beams. Optics express 18, 10777–10785 (2010).
 19.
Shu, W. et al. Polarization evolution of vector beams generated by qplates. Photonics Research 5, 64–72 (2017).
 20.
Bliokh, K. Y., RodrguezFortuño, F. J., Nori, F. & Zayats, A. V. Spin–orbit interactions of light. Nature Photonics 9, 796 (2015).
 21.
Cardano, F. & Marrucci, L. Spin–orbit photonics. Nature Photonics 9, 776 (2015).
 22.
Zhang, P. et al. Implementation of onedimensional quantum walks on spinorbital angular momentum space of photons. Physical Review A 81, 052322 (2010).
 23.
Cardano, F. et al. Quantum walks and wavepacket dynamics on a lattice with twisted photons. Science advances 1, e1500087 (2015).
 24.
Cardano, F. et al. Statistical moments of quantumwalk dynamics reveal topological quantum transitions. Nature communications 7, 11439 (2016).
 25.
Giordani, T. et al. Experimental engineering of arbitrary qudit states with discretetime quantum walks. Physical review letters 122, 020503 (2019).
 26.
Piccirillo, B., D’Ambrosio, V., Slussarenko, S., Marrucci, L. & Santamato, E. Photon spintoorbital angular momentum conversion via an electrically tunable qplate. Applied Physics Letters 97, 241104 (2010).
 27.
Karimi, E., Piccirillo, B., Nagali, E., Marrucci, L. & Santamato, E. Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned qplates. Applied Physics Letters 94, 231124 (2009).
 28.
El Ketara, M. & Brasselet, E. Selfinduced nonlinear spin–orbit interaction of light in liquid crystals. Optics letters 37, 602–604 (2012).
 29.
Barboza, R. et al. Vortex induction via anisotropy stabilized lightmatter interaction. Physical review letters 109, 143901 (2012).
 30.
Kravets, N., Podoliak, N., Kaczmarek, M. & Brasselet, E. Selfinduced liquid crystal qplate by photoelectric interface activation. Applied Physics Letters 114, 061101 (2019).
 31.
SánchezLópez, M. M., Abella, I., PuertoGarca, D., Davis, J. A. & Moreno, I. Spectral performance of a zeroorder liquidcrystal polymer commercial qplate for the generation of vector beams at different wavelengths. Optics & Laser Technology 106, 168–176 (2018).
 32.
Foo, G., Palacios, D. M. & Swartzlander, G. A. Optical vortex coronagraph. Optics letters 30, 3308–3310 (2005).
 33.
Serabyn, E., Mawet, D. & Burruss, R. An image of an exoplanet separated by two diffraction beamwidths from a star. Nature 464, 1018 (2010).
 34.
Nersisyan, S. R., Tabiryan, N. V., Mawet, D. & Serabyn, E. Improving vector vortex waveplates for highcontrast coronagraphy. Optics express 21, 8205–8213 (2013).
 35.
Aleksanyan, A., Kravets, N. & Brasselet, E. Multiplestar system adaptive vortex coronagraphy using a liquid crystal light valve. Physical review letters 118, 203902 (2017).
 36.
Pu, M. et al. Spatially and spectrally engineered spinorbit interaction for achromatic virtual shaping. Scientific Reports 5, 9822 (2015).
 37.
Radwell, N., Hawley, R., Götte, J. & FrankeArnold, S. Achromatic vector vortex beams from a glass cone. Nature communications 7, 10564 (2016).
 38.
Nassiri, M. G. & Brasselet, E. Multispectral management of the photon orbital angular momentum. Physical review letters 121, 213901 (2018).
 39.
Nassiri, M. G. & Brasselet, E. Pure and achromatic spinorbit shaping of light from fresnel reflection off spacevariant anisotropic media. Physical Review A 99, 013836 (2019).
 40.
Wakayama, T., Komaki, K., Otani, Y. & Yoshizawa, T. Achromatic axially symmetric wave plate. Optics express 20, 29260–29265 (2012).
 41.
Rafayelyan, M. & Brasselet, E. Braggberry mirrors: reflective broadband qplates. Optics letters 41, 3972–3975 (2016).
 42.
Gecevicius, M. et al. Toward the generation of broadband optical vortices: extending the spectral range of a qplate by polarizationselective filtering. JOSA B 35, 190–196 (2018).
 43.
 44.
Tabiryan, N., Nersisyan, S., Xianyu, H. & Serabyn, E. Fabricating vector vortex waveplates for coronagraphy. In 2012 IEEE Aerospace Conference, 1–12 (IEEE, 2012).
 45.
Tabirian, N., Xianyu, H. & Serabyn, E. Liquid crystal polymer vector vortex waveplates with submicrometer singularity. In 2015 IEEE Aerospace Conference, 1–10 (IEEE, 2015).
 46.
Pancharatnam, S. Achromatic combinations of birefringent plates parti. In Proceedings of the Indian Academy of SciencesSection A, vol. 41, 130–136 (Springer, 1955).
 47.
Pancharatnam, S. Achromatic combinations of birefringent plates partii. In Proceedings of the Indian Academy of SciencesSection A, vol. 41, 137–144 (Springer, 1955).
 48.
HerreraFernandez, J. M., Vilas, J. L., SanchezBrea, L. M. & Bernabeu, E. Design of superachromatic quarterwave retarders in a broad spectral range. Applied optics 54, 9758–9762 (2015).
 49.
Messaadi, A., SánchezLópez, M. M., Vargas, A., GarcaMartnez, P. & Moreno, I. Achromatic linear retarder with tunable retardance. Optics letters 43, 3277–3280 (2018).
 50.
Yi, X. et al. Addition and subtraction operation of optical orbital angular momentum with dielectric metasurfaces. Optics Communications 356, 456–462 (2015).
 51.
Delaney, S., SánchezLópez, M. M., Moreno, I. & Davis, J. A. Arithmetic with qplates. Applied optics 56, 596–600 (2017).
 52.
Tabiryan, N., Roberts, D., Steeves, D. & Kimball, B. New 4g optics technology extends limits to the extremes. Photonics Spectra 51, 46–50 (2017).
 53.
D’Ambrosio, V., Carvacho, G., Agresti, I., Marrucci, L. & Sciarrino, F. Tunable twophoton quantum interference of structured light. Physical Review Letters 122, 013601 (2019).
 54.
Iizuka, K. Elements of Photonics, Volume I: In Free Space and Special Media, vol. 1 (John Wiley& Sons, 2002).
 55.
Born, M. & Wolf, E. Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).
 56.
Bhandari, R. & Love, G. D. Polarization eigenmodes of a qhq retarder—some new features. Optics communications 110, 479–484 (1994).
 57.
Shu, W. et al. Propagation model for vector beams generated by metasurfaces. Optics express 24, 21177–21189 (2016).
 58.
Ji, W. et al. Metaqplate for complex beam shaping. Scientific reports 6, 25528 (2016).
 59.
Collett, E. Field guide to polarization. (Spie Bellingham, WA, 2005).
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G.K. had the initial theoretical ideas towards realizing tunable retardance qplates and has carried out relavant simulations. Examining these results for wavelengthindependence was carried out by G.R. Analytical calculations and experiments were performed by R.B. All three authors have contributed equally towards analyzing the results and in manuscript writing.
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Correspondence to Gururaj Kadiri.
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B, R., Kadiri, G. & Raghavan, G. Wavelengthadaptable effective qplates with passively tunable retardance. Sci Rep 9, 11911 (2019) doi:10.1038/s41598019481638
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