Pancharatnam-Berry optical element sorter of full angular momentum eigenstate

Gary F. Walsh

doi:10.1364/OE.24.006689

1. Introduction

Ever increasing bandwidth demands and power consumption in telecommunication systems has attracted research into new ways to maximize information transmission rates while minimizing energy consumption. Engineers have encoded data in the amplitude, phase, polarization, wavelength, and time signature of optical waves leaving space as the last degree of freedom to exploit. One particularly interesting concept in this area is to modulate orbital angular momentum (OAM) [1–5]. Due to the infinite number of orthogonal OAM eigenstates there is in principle no limit to the amount of information that a single photon can carry [6]. Additionally, OAM entanglement in nonlinear optical processes can enable high bandwidth unbreakable quantum cryptography [7]. In order to take advantage of these opportunities, efficient, fast, and cost effective OAM modulators and sorters must be developed. These devices need to be able to manipulate precisely the phase front of optical waves. The first demonstration of the ability to sort OAM for a single photon used a cascading series of Mach-Zehnder interferometers [1]. This implementation was large and expensive making it impractical for telecommunication applications. Recently a highly efficient OAM sorting method was developed utilizing an optical geometric transform that converts helical phase into a linear phase gradient, and was demonstrated with both diffractive [8] and refractive [9] elements. This scheme solved the size issue, requiring only two complex elements in a 4-f imaging system; however, these elements must be individually diamond machined making them expensive and limiting their effectiveness for practical applications.

One possible solution is to use Pancharatnam-Berry optical elements (PBOE) [10,11]. PBOEs are anisotropic thin planar films or structures that induce a transversely nonuniform topological phase transformation resulting in an overall wavefront reshaping for a transmitted optical beam. These elements have been realized through a variety of methods including liquid crystal [12,13] and polymer liquid crystal diffractive waveplates [14–18], metasurfaces composed of subwavelength dielectric gratings [11,19–21], and V-shaped plasmonic antennas [22,23]. Unlike traditional diffraction gratings, PBOEs have been shown to produce near 100% diffraction efficiency for a single non-zero diffraction order, enabling ultra-thin, light-weight elements that perform similar functions as bulk refractive optics and have in addition the inherent ability to selectively act on spin angular momentum (SAM) eigenstates. Furthermore, a single beam holographic replication process for polymer liquid crystal diffractive waveplate PBOEs has been demonstrated using photo-isomerizing dyes and ultraviolet polymerization of liquid crystal monomer precursor [16]. PBOEs manufactured with these materials have obtained resolution of less than 3μm [15], which is better than what would be required to produce the devices proposed in this work. This development suggests that PBOEs could soon become manufacturable as a thin lightweight alternative to traditional optics.

In this paper, using an analytic simulation method, we design the two-dimensional topological phase transformations of two PBOEs that simultaneously sort the OAM, and SAM eigenstate composition of any optical beam using a 4-f optical system. We show that this device uniquely identifies both the OAM and SAM components of Gaussian beams modulated with helical phase fronts, as well as Laguerre-Gaussian, Hermite-Gaussian, and higher order and mixed paraxial laser cavity modes. We show that this device can decompose structured polarization fields to characterize fully their component OAM eigenmodes. If realized these elements could significantly reduce the size and cost of OAM sorters for communication systems, quantum-cryptography, laser and fiber modal output characterization, and scientific measurement techniques such as the full characterization of the high dimensional OAM state vector [24].

2. Pancharatnam-Berry angular momentum sorter

Similar to elementary particles, photons possess two types of quantized angular momentum. SAM, commonly referred to as polarization, has two orthogonal eigenstates, namely, left-circular polarization $| L 〉$ , and right-circular polarization $| R 〉$ . Left and right refer to the direction in which the electric field vector rotates as the wave propagates. Additionally photons possess quantized values of OAM with a boundless set of possible eigenstates. OAM refers to the azimuthal phase of the wave. A beam with a topological phase dependence of $\exp (j l ϕ)$ , where $ϕ$ is the polar angle on a plane perpendicular to the propagation direction, carries OAM of $l ℏ$ that is independent of the polarization. The integer l is known as the topological charge. Beams with $l \neq 0$ are commonly referred to as vortex beams because the phase gradient wraps around a central singularity creating an intensity null similar to vortices encountered in other areas of physics.

PBOEs are patterned half-waveplates that modulate the Pancharatnam (topological) phase of a beam by transversely varying the polarization rotation. A standard half-waveplate consists of an aligned anisotropic material, which rotates the linear polarization vector by an angle $2 α$ , where $α$ is the angle between the slow-axis of the material and the polarization vector. When acting on a circularly polarized wave this element inverts the SAM state. This mapping is mathematically represented with a rotation Jones matrix. If instead of being fixed at a constant angle, the slow axis orientation follows a transversely varying two-dimensional function $α (x, y)$ , then the polarization rotation changes across the wavefront. The two-dimensional polarization transform of PBOEs are characterized by a spatially varying rotation Jones matrix [10],

M (x, y) = [\begin{matrix} \cos 2 α (x, y) & \sin 2 α (x, y) \\ \sin 2 α (x, y) & - \cos 2 α (x, y) \end{matrix}] .

Applying this matrix to an input wave in the $| L 〉$ SAM eigenstate, ${\overset{⇀}{E}}_{0} = [1; j]$ , the output becomes $| R 〉$ and the wave front is modulated by the topological phase factor $\exp (j 2 α (x, y))$ , e.g. $\overset{⇀}{E} = E_{0} M (x, y) [1; j] = E_{0} \exp (j 2 α (x, y)) [1; - j]$ . Conversely, for a $| R 〉$ input the output is $| L 〉$ with a topological phase modulation $\exp (- j 2 α (x, y))$ . PBOEs are therefore thin, phase-only optical elements which act separately on the two SAM eigenstates. This property allows us to construct an optical system that simultaneously sorts the OAM and SAM of an optical beam.

Berkhout et al. demonstrated that spatially separating orthogonal OAM states can be accomplished by mapping the azimuthal phase of a vortex beam into a linear phase gradient [8]. This transforms OAM into linear momentum producing a displacement of the beam when focused to a spot in an image plane. The topological charge of the vortex beam determines the spatial frequency of the linear phase gradient and therefore the spatial separation of the sorted OAM states. This process is implemented with a ring-to-point geometric optical transformation using two complex optical elements in a 4-f imaging system. In this section, we show that this transformation can be implemented using PBOEs, and can additionally be modified to simultaneously sort the SAM eigenstates taking advantage of their inherent circular polarization selective properties.

The ring-to-point geometric optical transform that sorts OAM maps $(x, y) \to (u, v)$ , where $(x, y)$ and $(u, v) = (a \tan^{- 1} (y / x), - a \ln (\sqrt{x^{2} + y^{2}} / b))$ are the Cartesian coordinates of the input and output planes, respectively [8, 25]. The variables $a$ and $b$ scale and translate the transformed image in the Fourier plane of the 4-f system. The choice of $a$ is based on the desired size and spread of the sorted spots in the final image as discussed in [8], Berkhout et al. b is chosen such that the second element may be a practical size. For example, in this paper we use numerical calculations with a long focal length and choose b = 1 while in [9], Lavery et al, where this transform was implemented with refractive elements, the authors used a much shorter focal length producing a larger spread of the beam in the Fourier plane. They chose b = 0.00447 to translate the image in the Fourier plane back near the center of the second element so that it could be fabricated on a standard table top optic. The derivation of the holographic phase masks that implement the ring-to-point geometric transformation is given in [25], Cederquist et al. The authors use the method of stationary phase to derive a general approximation for the two-dimensional phase of any conformal holographic mapping. Solving for a series of two conformal mappings gives the solution for the non-conformal ring-to-point transform. This transformation can be implemented using PBOEs, which can additionally be adapted to simultaneously sort the SAM eigenstates. The two PBOEs are referred to as the transformation and phase correction elements and their transversely varying slow-axis orientation patterns are given by,

α_{1} (x, y) = \frac{π a}{λ f} (y \tan^{- 1} (\frac{y}{x}) - x \ln (\frac{\sqrt{x^{2} + y^{2}}}{b}) + x),

and

α_{2} (x, y) = \frac{π}{Λ} x + \frac{π a b}{λ f} {\begin{matrix} (- 1) \exp (- \frac{x}{a}) \begin{matrix} \cos (\frac{y}{a}); & x \geq 0 \end{matrix} \\ (+ 1) \begin{matrix} \exp (+ \frac{x}{a}) \cos (\frac{y}{a}); & x < 0 \end{matrix} \end{matrix},

respectively, where f is the focal length of the lenses in the 4-f imaging system, and determines the spacing between the sorted SAM eigenstates. Figures 1(a) and 1(b) show an example of

α_{1} (x, y)

, and

α_{2} (x, y)

. Figures 1(c) and 1(d) show the director patterns of the corresponding PBOEs, where each line segment indicates the local orientation of the slow axis.

Fig. 1 The angle $α (x, y)$ of an implementation of the (a) transformation $α_{1} (x, y)$ and (b) phase correction $α_{2} (x, y)$ PBOEs. Two-dimensional slow-axis orientation map of the (c) transformation and (d) phase correction PBOEs.

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Figure 2(a) shows the proposed sorting device, a 4-f imaging system with the transformation and phase correction PBOEs placed in the object and Fourier planes. Notice the difference in the path that the $| L 〉$ and $| R 〉$ SAM states of the input beam take through the system. This occurs because in each definition of the PBOE orientation patterns, Eq. (2) and Eq. (3), there is a term that is linearly proportional to the x-dimension. This is the phase factor of a cycloidal PBOE, which acts as a diffraction grating that perfectly couples $| L 〉$ to the + 1 and $| R 〉$ to the −1 diffraction orders [12,14]. The term $x (π a / λ f)$ in Eq. (2) comes out of the derivation of the ring-to-point geometric transformation [25]. Due to this term, the two SAM states are spatially separated in the Fourier plane allowing the phase correction PBOE to operate independently on each. We are then able to properly account for the inherent difference in the way PBOEs act on opposite circular polarizations. Recall that the phase modulation of a PBOE on the two SAM states have opposite signs, and that after transmission through each PBOE the SAM states are inverted $(| L 〉 \to | R 〉, | R 〉 \to | L 〉)$ . We account for these differences with the piecewise definition of $α_{2} (x, y)$ in Eq. (3) with sign changes on either side of the x-axis. Without the spatial separation in the Fourier plane, PBOEs would not be able to properly modulate the phase of both SAM states. Finally, in order to separate the SAM states in the sorted image, we superimpose the phase factor of second cycloidal PBOE on the phase correction element of the ring-to-point transformation by adding the term $x π / Λ$ in Eq. (3). This additional mapping has previously been used to implement a single diffractive element measurement of circular dichroism [26]. Integration of this term into the phase correction element adds this capability to what would otherwise be an OAM only sorter. By removing SAM degeneracy this doubles the available state space for quantum communication and gives the device the ability to fully characterize the fractional contribution of each SAM and OAM eigenstate to the total field. Without this term $(\pm l ℏ, | L 〉)$ and $(\mp l ℏ, | R 〉)$ would appear in the same location in the final image. In that case, for example, beam power in the left-circularly polarized $+ 1 ℏ$ OAM eigenstate could not be distinguished from that of the right-circularly polarized $- 1 ℏ$ state.

Fig. 2 (a) The OAM and SAM sorter, a 4-f imaging system with a transformation PBOE in the Object plane, and phase correction PBOE in the Fourier plane. The sorted angular momentum eigenstates appear in the Image plane. (b) Optical intensity of a linearly polarized input Gaussian beam modulated by a topological phase factor $\exp (j 2 l ϕ)$ than propagated 500mm to the transformation PBOE with OAM $2 ℏ$ , $λ$ = 1550nm. (c) Intensity distribution at the Transform plane. The $| L 〉$ SAM component is imaged onto the –x side of the Transform plane while the $| R 〉$ SAM component is imaged onto the + x side, allowing the phase correction PBOE to independently modulate the two SAM states. (d) Intensity at the Image plane. $| L 〉$ SAM is focused to a point onto the + x side and $| R 〉$ SAM onto the -x. The focused spots are displaced in the y-direction according to the OAM eigenstate. Sorter parameters are $a$ = 1.3mm, b = 1, $Λ$ = 1.8mm, and f = 500mm. All intensity profiles are normalized to their own maximum intensity.

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Figure 2(b) shows the intensity profile at the input of the transformation PBOE for a Gaussian beam modulated by a topological phase factor $\exp (j 2 ϕ)$ , giving it an OAM of $+ 2 ℏ$ . The phase factor is inscribed at the beam waist. The transformation element is located 500mm from that point. The beam has a waist radius of $W_{0}$ = 5mm, a wavelength of $λ$ = 1550nm, and is linearly polarized. Notice the characteristic intensity null in the center of the beam resulting from the phase singularity. Figure 2(c) shows the intensity profile of this beam at the Fourier plane where the phase correction PBOE is applied. The $| L 〉$ SAM component appears in the –x side of the plane while the $| R 〉$ component appears in the + x side. As mentioned previously, this allows the phase correction PBOE to modulate independently the two SAM eigenstates. The role of the phase correction element is to compensate for phase distortions produced by the transformation element and is required for any non-conformal optical mapping [27]. When defining the orientation function for the PBOE implementation of this element we must be careful to account properly for the inversion of the SAM states and the discrepancy in the sign of the phase modulation.

Figure 2(d) shows the sorted OAM and SAM eigenstates of this beam. $| L 〉$ is focused onto the + x side of the image plane while $| R 〉$ is focused onto the –x side. The focused spots are displaced in the y-direction by a distance determined by their OAM eigenstates in accordance with the ring-to-point geometric optical transform. For $| L 〉$ the OAM displacement has the same sign as the topological charge while it has the opposite sign for $| R 〉$ .

The intensity patterns in Figs. 2(b)-2(d) were calculated using fast Fourier transforms to solve for the paraxial propagation between elements by applying a diffraction operator $\hat{D} = (1 / 2 j k_{0}) \nabla^{2}$ to the complex amplitude of the beam [28]. Assuming the space between the elements is isotropic this operator may be applied separately to the x and y vector components of the field. Transmission through each lens is calculated by multiplying by a transverse phase function $E_{x, y}^{+} (x, y) = \exp (j k_{o} (x {}^{2}+ y^{2}) / 2 f) E_{x, y}^{-} (x, y)$ , and through the transformation and phase correction PBOEs by applying the spatially varying rotation Jones matrix of Eq. (1) $[E_{x}^{+} (x, y); E_{y}^{+} (x, y)] = M (x, y) [E_{x}^{-} (x, y); E_{y}^{-} (x, y)]$ , where f is the focal length of the lenses, $k_{0}$ is he wavenumber, and $E_{x, y}^{-}$ and $E_{x, y}^{+}$ are the vector field components of the beam as it enters, and exits each element. Since the lenses are isotropic the phase function may be applied separately to each vector field component while the Jone’s matrix must be used for the PBOEs to account for changes in the polarization. The parameters of the sorter are $a$ = 1.3mm, b = 1, $Λ$ = 1.78mm, and f = 500mm.

Figure 3 illustrates the angular momentum sorting behavior of this device for Gaussian beams modulated with varying topological charge. Figure 3(a) shows a superposition of the intensity profiles in the output plane, for incident beams with discrete OAM values between $- 12 ℏ$ to $12 ℏ$ , for both SAM eigenstates. In a quantum OAM communication system, the number of states in Fig. 3(a) would have a channel capacity to decode 3.73 bits per photon compared with the theoretical limit of 4.17, which was calculated following the mutual information method from [29]. The step of 3 $ℏ$ was chosen to reduce cross-talk between channels. Figure 3(b) shows the y-positions of each sorted OAM state from $- 16 ℏ$ to $16 ℏ$ . The spacing is linear with the slope following the trend reported in [8], Berkhout et al $Δ y = 2 π λ f / a$ . If each of these states could be completely discriminated the sorter could decode 6.04 bits per photon. However, as can be seen in Fig. 3(c) which shows the intensity cross-section along the y-direction for l = 0 to 4, the sorter output for adjacent states overlap around 80% of their maxima leading to a channel capacity of only 5.21 bits per photon for the 66 angular momentum states in Fig. 3(b). In order to explore the causes of cross-talk in the design of the optical system we performed a multi-variable study of the sorter parameters. We varied f, $λ$ , and b, while holding $a$ , w_o, and the distance between the beam waist and the transform element constant. We empirically found that the full width half maximum spread of the focused spots in the y-direction averaged between l = 0 to 4 is fwhm = $231 λ f$ , with the slope having the units of m⁻¹ and the regression fitting with R² > 0.999. For comparison, for this value of $a$ , $Δ y = 125 λ f$ . That is the spread of the points in the sorter output are 85% larger than the spacing between them no matter the choice of wavelength or focal length. It was found that b impacts neither $Δ y$ nor fwhm. In order to maximize channel capacity an additional strategy must be taken beyond the tuning of the sorter parameters, for instance reducing overlap of the sorted states has been demonstrated by refractive beam copying [29,30]. This may be implemented by adding additional PBOEs to the system. The ability of these ultra-thin elements to uniquely identify a large number of OAM and both SAM eigenstates in a single measurement suggests the impact that PBOEs could have in highbandwidth communication and encryption systems. However, in order for this potential to be realized these properties must be maintained when sorting laser cavity and fiber modes, which is the subject of the next section.

Fig. 3 (a) A superposition of the sorted OAM eigenstates from a Gaussian beam modulated by a phase factor $\exp (j l ϕ)$ for l = 12, 9, 6, 3, 0, −3, −6, −9, −12 for both left-circular and right-circular polarization. The intensity of each sorter output is normalized to its own maximum. (b) y-position of each sorted OAM and SAM eigenstate. (c) Cross-section along the y-direction of the sorter output for l = 0 – 4.

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3. Sorting angular momentum of laser cavity modes

Up to this point we have considered the sorting operation of the proposed device for zero-order Gaussian beams modulated by helical phase factors. This type of OAM carrying beam is the type generated by a q-plate [31]. Alternatively, laser and fiber [32] modes carrying OAM can also be switched at high rates for use in data transmission. A particularly important set of cavity modes are the Laguerre-Gaussian modes, which are a solution to the paraxial Helmholtz equation, and are a complete set of OAM eigenmodes. The complex amplitude of the Laguerre-Gaussian mode of order (l,m) at the beam waist is,

L G_{m}^{l} (x, y; z = 0) = {(\frac{| \sqrt{x^{2} + y^{2}} |}{W_{o}})}^{l} L_{m}^{l} (\frac{2 (x^{2} + y^{2})}{W_{0}^{2}}) \exp (- \frac{(x^{2} + y^{2})}{W_{0}^{2}} - j l φ),

where

L_{m}^{l} (w)

are the generalized Laguerre polynomials defined using he Rodrigues formula [31]. l is still the topological charge and can be any integer, while m is an integer

\geq

0. Similar to the intensity profile in Fig. 2(b), for l $\neq$ 0 the field of the Laguerre-Gaussian modes has a central singularity resulting in an intensity null. The intensity profile of the

L G_{0}^{+ 1}

, and

L G_{0}^{- 1}

modes are shown in Fig. 4(a) and 4(b), respectively, for W₀ = 5mm at z = 0. These intensity profiles are identical, and therefore the sign of the topological charge cannot be determined without an additional phase measurement.

Fig. 4 Intensity profile of the (a) $L G_{0}^{+ 1}$ and (b) $L G_{0}^{- 1}$ Laguerre-Gaussian modes, and the (c) $H G_{01} = L G_{0}^{+ 1} + L G_{0}^{- 1}$ Hermite-Gaussian mode. Sorter output for the $L G_{0}^{+ 1}$ mode (a1) left-circularly polarized, (a2) right-circularly polarized, and (a3) linearlly polarized. (b1), (b2), and (b3) the same for $L G_{0}^{- 1}$ . (c1), (c2), and (c3) the same for $H G_{01}$ . All intensity profiles are normalized to their own maximum.

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The two SAM eigenstates of a Laguerre-Gaussian mode are defined as $| L 〉 = L G_{m}^{l} \hat{x} + j L G_{m}^{l} \hat{y}$ , and $| R 〉 = L G_{m}^{l} \hat{x} - j L G_{m}^{l} \hat{y}$ . For the $L G_{0}^{+ 1}$ and $L G_{0}^{- 1}$ modes, the outputs of the proposed sorting device are shown in Figs. 4(a1) and 4(b1) for $| L 〉$ , and Figs. 4(a2) and 4(b2) for $| R 〉$ . Although the intensity profile of the four input beams are identical, one non-destructive measurement using the proposed sorter can simultaneously determine both the OAM and SAM eigenstate.

The sorter can also determine if the beam is in a mixed SAM state such as linear or elliptical polarization. Figures 4(a3) and 4(b3) show the sorting results for one such linearly polarized state, $L P = L G_{m}^{l} \hat{x} + L G_{m}^{l} \hat{y}$ , for l = + 1 and −1 respectively. The input intensity profiles are identical to those of the four eigenstate pairs. However, since the mixed SAM states are superpositions of the $| L 〉$ and $| R 〉$ , the output of the sorter shows the relative intensity of each component. By measuring the intensity difference between the spots on either sides of the y-axis, the contribution of each SAM eigenstate to the total field can be determined. However, this measurement will not determine the phase difference between the modes. For example, the results for all linearly polarized $L G_{0}^{+ 1}$ and $L G_{0}^{- 1}$ modes will be identical to Figs. 4(a3) and 4(b3), respectively, no matter the polarization angle.

Another important set of cavity modes are the Hermite-Gaussian modes. The complex amplitudes of which are defined by

H G_{p q} (x, y) = W_{o}^{p + q} \frac{\partial^{p}}{\partial x^{p} \partial y^{q}} G_{0} (x, y),

where p and q are integers, and G₀ is the complex amplitude of the zero-order Gaussian beam [33]. Equation (5) can be reconstructed from a linear superposition of phase delayed

L G_{m}^{\pm l}

modes [34]. An example of this is illustrated in Fig. 4(c), which shows the intensity of the mode

H G_{01} = L G_{0}^{+ 1} + L G_{0}^{- 1}

. As was the case for the pure Laguerre-Gaussian eigenmodes, these mixed OAM modes can exist in any SAM state. Figures 4(c1) and 4(c2) show the output of the sorter for the HG₀₁ mode in

| L 〉

and

| R 〉

, respectively. The two focused spots in each figure correspond to the

+ 1 ℏ

and

- 1 ℏ

OAM eigenstates. Since these spots appear on one side of the image plane in the x-direction, only one SAM eigenstate is present in each. Analogous to the measurement of the contribution of each SAM eigenstate, the fraction of each

L G_{m}^{l}

OAM eigenmode but not their phase difference can be determined by measuring the relative intensity of the spots on either side of the x-axis. An equivalent interpretation of this result is that this measurement will determine the phase difference between the HG_pq and HG_qp modes but not their fractional contribution to the field. Figure 4(c3) shows the output for a linearly polarized version of this Hermite-Gaussian mode. In this case all four first order OAM and SAM eigenstates are present. For all linearly polarized first order Hermite-Gaussian modes the sorter output will be identical to Fig. 4(c3), no matter the polarization or rotation angle. However, the sorter can determine the fractional contribution of each Laguerre-Gaussian eigenmode of a complex mixed modal beam.

Figure 5(a) illustrates the concept of mixed SAM states of a multi-modal beam. The Poincaré-sphere is a tool for visualizing pure states of polarization. The SAM eigenstates are represented by the north and south poles and the equator depicts all possible linear polarization states. The s and p linear polarization states are at the front and rear of the equator. The elevation angle determines the relative contribution of the two eigenstates and the azimuth describes the phase delay between them. States not on the equator or at the poles are elliptically polarized. Monitoring the relative intensity of the focused spots on either side of the y-axis of the PBOE angular momentum sorter output will determine the elevation angle but not the azimuth. That is, it finds the relative magnitude of each eigenstate but not their phase difference.

Fig. 5 (a) Poincaré-sphere. (b) Fraction of total power in the $| L 〉$ (blue) and $| R 〉$ (red) spots in the PBOE angular momentum sorter output. (c) Equivalent Poincaré-sphere for the $L G_{0}^{\pm 1}$ modes. (d) Fraction of total power in the $L G_{0}^{+ 1}$ (blue) and $L G_{0}^{- 1}$ (red) spots in the PBOE angular momentum sorter output.

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We define the complex amplitude of a mixed SAM state Laguerre-Gaussian mode as,

[\begin{matrix} U_{x} \\ U_{y} \end{matrix}] = [\begin{matrix} L G_{m}^{l} \\ L G_{m}^{l} \exp (j (θ + \frac{π}{2})) \end{matrix}],

where

θ

is the elevation angle in Fig. 5(a). For the l = + 1, m = 0 modes the sorter output for the

| L 〉

and

| R 〉

SAM eigenstates are shown in Figs. 4(a1) and 4(a2). Figure 5(b) shows the fraction of total output power in each of these two spots as a function of

θ

. The two curves have a sinusoidal shape. Using these curves the elevation angle on the Poincaré-sphere of an elliptically polarized beam and thereby the fractional contribution of each SAM eigenstates to the total field may be experimentally determined. This function could be used in sensitive polarimetry techniques such as ellipsometry.

The same technique can also be used to measure the relative contribution of each OAM eigenstate to the total field. Figure 5(c) shows the equivalent Poincaré-sphere that was introduced by Padgett and Courtail for visualizing the OAM components of these modes [34]. This sphere specifically illustrates the mixing of the l = $\pm 1$ OAM eigenstates. In this representation a separate equivalent Poincaré-sphere exists for each pair l = $\pm Q$ , where $Q$ is an integer > 0. Similar to the Poincaré-sphere, the poles represent the eigenmodes, which in this case are $L G_{0}^{+ 1}$ and $L G_{0}^{- 1}$ . The equator depicts all possible equal intensity linear combinations of $L G_{0}^{+ 1}$ and $L G_{0}^{- 1}$ , or equivalently all equal phase linear combinations of HG₁₀ and HG₀₁. By analogy with the SAM Poincaré-sphere we see that HG₁₀ and HG₀₁ are the OAM equivalents of s and p polarization. For $| L 〉$ , we define the complex amplitude of a mixed OAM state Laguerre-Gaussian mode as,

[\begin{matrix} U_{x} \\ U_{y} \end{matrix}] = [\begin{matrix} L G_{m}^{+ | l |} \cos (\frac{θ}{2}) - L G_{m}^{- | l |} \sin (\frac{θ}{2}) \\ j L G_{m}^{+ | l |} \cos (\frac{θ}{2}) - j L G_{m}^{- | l |} \sin (\frac{θ}{2}) \end{matrix}],

where is the elevation angle in Fig. 5(c). For m = 0, the sorter output for the l = + 1 and −1 eigenmodes are shown in Figs. 4(a1) and 4(b2) for

| L 〉

. Figure 5(d) shows the fraction of total output power in each of these two spots as a function of the elevation angle on the equivalent Poincaré-sphere. The two curves are identical to those in Fig. 5(b). We can therefore use the same method to measure the relative contribution of each OAM eigenmode to the total field. As will be discussed in greater detail in the next section, this function could be used to characterize the modal contributions of fiber optic outputs either for modal quality measurements or high rate data transmission.

In this section, we have demonstrated the ability of the proposed PBOE sorting device to determine the angular momentum eigenstates of Laguerre-Gaussian, and Hermite-Gaussian laser cavity modes. Thus demonstrating its possible implementation as a decoder in communications systems where OAM and SAM eigenstates are switched within a laser cavity or fiber, or as an evaluation tool to characterize the modal composition of lasers and fibers. In the next section, we examine the ability of the sorter to decode information from higher order and structured hybrid modes.

4. Sorting higher order and hybrid, laser cavity modes

In the previous section, we evaluated the performance of the proposed PBOE angular momentum sorter for Laguerre-Gaussian cavity eigenmodes of order m = 0 as well as Hermite-Gaussian modes, and for characterization of laser and fiber modal composition. For this device to be an effective tool for high bandwidth communication systems utilizing mode division multiplexing [35], this analysis must be expanded to include higher order and hybrid modes which possess a more complex combination of SAM and OAM eigenstates. A common example of hybrid modes are the radially polarized transverse magnetic $T M_{01} = \pm [H G_{10}; H G_{01}]$ , and azimuthally polarized transverse electric $T M_{10} = \pm [H G_{10}; - H G_{01}]$ vector modes [36]. These can be constructed from the superposition of two Laguerre-Gaussian eigenmodes, one in each SAM eigenstate, and have the same structured polarization patterns as two of the four first order fiber optics eigenmodes [37]. However, unlike the Hermite-Gaussian modes, the component Laguerre-Gaussian modes are in opposite OAM eigenstates, and are rotated from one another by 90°. This gives them intensity patterns identical to the pure Laguerre-Gaussian OAM eigenmodes, but with more complex angular momentum structure, which is evidenced in their nonuniform polarization patterns³⁵. Figures 6(a) and 6(b) show the intensity and polarization patterns of the TM₀₁, and TE₀₁ vector modes, respectively. The intensity patterns are normalized to their own maximum, and the local polarization directions are depicted by the arrows. Notice that these beams have identical intensity patterns to the $L G_{0}^{\pm 1}$ modes in Figs. 4(a) and 4(b); however, their polarization differs from the trivial circular, linear, and elliptical states of the pure eigenmodes. The electric field vectors of these hybrid modes are linearly polarized in a direction that varies transversely across the beam. Note that despite the central null in the beam intensity these vector modes have no net OAM, similar to the Hermite-Gaussian modes. That is, they are equal parts $\pm 1 ℏ$ with the same probability of measuring either eigenvalue [38]. This can be seen in Fig. 6(c) which shows the output of the sorter for both the TM₀₁ and TE₀₁ vector modes. Notice that the eigenstate composition of both of these structured beams is equal magnitude $(| L 〉, L G_{0}^{+ 1})$ and $(| R 〉, L G_{0}^{- 1})$ . This can be verified by comparing these results to the definitions. The sorter cannot determine the difference between these modes since they have the same eigenstate composition. However, two other vector modes exist with the same polarization structure as the two remaining first order fiber optic eigenmodes, specifically, hybrid electric odd HE_21o = $\pm$ [HG₀₁; HG₁₀], and hybrid electric even HE_21e = $\pm$ [HG₀₁; -HG₁₀]. Figures 6(d) and 6(e) show the intensity and polarization patterns of the HE_21o, and HE_21e vector modes, respectively. Figure 6(f) shows the sorter output for both of these beams. It now becomes clear that the angular momentum eigenstate pairs of the HE_21o, and HE_21e vector modes are $(| L 〉, L G_{0}^{- 1})$ and $(| R 〉, L G_{0}^{+ 1})$ , the opposite of TM₀₁ and TE₀₁.

Fig. 6 The intensity profile and structured transverse polarization direction of the (a) TM₀₁, (b) TE₀₁, (d) HE_21o, and (e) HE₂₁_e vector beams. Sorter results for (c) the TM₀₁ and TE₀₁, and (f) the HE_21o and HE_21e beams. All intensity profiles are normalized to their own maximum.

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The sorter can therefore in a single measurement, decompose the angular momentum eigenmode components of structured polarization hybrid beams, which cannot be simultaneously measured using other methods. For example, an OAM only sorter would not be able to distinguish the ring vector beams from a Hermite-Gaussian beam without making an additional measurement of the intensity pattern. In any case, it could not distinguish TM₀₁ / TE₀₁ from HE_21o / HE_21e as they all have equal parts $\pm 1 ℏ$ OAM. The structured polarization pattern could be measured directly by taking a series of images of the intensity pattern through a linear polarizer rotated at various angles [32]. The required resolution of the rotation angle will increase with the OAM eigenvalue. Recently, vector mode demultiplexing of all four vortex ring modes was demonstrated using a beam splitter and two q-plates of opposite sign charge to transform TM₀₁ / HE_21e, and TE₀₁ / HE_21o to s and p polarized zero order beams [35]. This allows for the simultaneous complete discrimination of the first order fiber vector modes but would be impractical to scale up since the number of elements and the insertion losses will increase linearly with the number of modes. Finally, a holographic interferometric method could be used, where the beam is combined with a reference, and a numerical image processing method decomposes the interference pattern into component Bessel beams [39], or a basis set of fiber vector modes [40]. None of these alternative methods possess the ability of the PBOE angular momentum sorter to characterize the eigenmode composition of a structured optical beam in a single intensity measurement without the need for interferometers and numerical image processing. The ability of the proposed device to fully discriminate the angular momentum eigenstates of hybrid modes is evidence of its versatility. The result that the device can operate for complex polarization structures indicates that the angular momentum eigenstates of any hybrid mode can be determined. This provides a vast available state space for communication systems. For example, with both SAM, and only two OAM eigenstates, there are 16 unique outputs which can encode 4 bits in a many photon non-quantum scheme. Adding the zero order Gaussian mode increases the capacity to 6 bits. All that is needed is the ability to form any linear combination of the $L G_{0}^{\pm 1}$ eigenmodes. This can be accomplished in cavities, free space, and fibers.

Another important class of higher order modes are $L G_{m}^{l}$ for m > 0. Similar to the m = 0 $L G_{m}^{l}$ modes, these exist in single OAM eigenstates, but have larger cross-sections consisting of series of concentric rings in intensity. Despite the OAM degeneracy of these modes, they also can be distinguished by the proposed PBOE sorter. Figures 7(a)-7(c) shows the intensity profile of the $L G_{m}^{+ 1}$ modes for m = 0, 1, and 2, respectively. Other than m all beam parameters are identical to those of the Laguerre-Gaussian modes discussed earlier, the intensity of each image is normalized to its own maximum. Notice that each mode pattern consists of m + 1 concentric rings. Figure 7(d) shows the intensity profile along the x-direction of the output of the sorter at the y-offset associated with the l = + 1 OAM eigenstate. Each of the cross-sections (m = 0 blue, m = 1 red, and m = 2 green) is normalized to its own maximum. The inset shows the two-dimensional intensity maps of each of these outputs. Notice that there are N = m + 1 intensity spots displaced in the x-direction of the sorted image, which is perpendicular to the off-sets produced by the OAM sorting operation. This result is in fact the original function of the “ring-to-point” coordinate transformation that is used to implement the azimuthal to linear phase mapping for sorting OAM²⁴. Importantly, the displacement in the y-direction does not change with m since this parameter does not alter the OAM eigenstate of the higher order Laguerre-Gaussian modes. This dual functionality gives the device additional ability to evaluate the modal composition of laser cavities and fibers. It is worth pointing out that for the current set of parameters the overlap between focused points in the sorted image increased for larger values of m, making it difficult to determine the exact modal composition of the beam. For example if both the $L G_{0}^{+ 1}$ , and $L G_{2}^{+ 1}$ modes were present in the laser output, determining the relative magnitude of each would require careful analysis of the line shape as both modes contribute power to the central peak. However, it should be possible to overcome these limitations with proper experimental design, for example reducing overlap through refractive beam copying [29,30].

Fig. 7 Intensity profile for $L G_{0}^{+ 1}$ (a), $L G_{1}^{+ 1}$ (b), and $L G_{2}^{+ 1}$ (c). (d) Intensity cross-sections of the sorter output along the + 1 OAM y-offset for $L G_{0}^{+ 1}$ (blue), $L G_{1}^{+ 1}$ (red), and $L G_{2}^{+ 1}$ (green). The inset shows the two-dimensional intensity profiles of these results. Each intensity profile and cross-section is normalized to the maximum value of the intensity profile. The sorter parameter $Λ$ was decreased to $444 μ m$ to increase separation between SAM states.

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Lastly, we provide an example pulling together the entire range of functionalities of the device including discrimination of OAM, SAM, the m parameter, and determination of the relative contribution of each cavity eigenmode. Figure 8 shows several complex structured beams composed of a superposition of Laguerre-Gaussian modes with l = −3, + 1, and + 5 for $| R 〉$ , and l = + 3, −1, and −5 for $| L 〉$ . In each the l = $\pm$ 1 modes contribute twice the power to the total field as the other modes, and the $| L 〉$ , and $| R 〉$ modes are rotated from each other by 90°. In Fig. 8(a1), the parameter m = 1 for all the component modes. The intensity pattern is normalized to its own maximum, and the local polarization direction is depicted by the arrows. Figure 8(a2) shows the sorter output for this structured field. Each of the Laguerre-Gaussian cavity mode components of the field are uniquely identified by the sorter. Notice that there are two focal spots for each OAM, SAM pair indicating that the m parameter of the eigenmodes is 1. Figures 8(b1), and 8(c1) show the structured beam where m has been changed to 0 for $| R 〉$ , and $| L 〉$ , respectively. Notice that the intensity cross-sections of these beams are identical and that their structured polarization patterns are nearly the same, the only difference being that they are a half-wave out of phase. This subtle difference is detected by the PBOE sorter, as can be seen by comparing Figs. 8(b2), and 8(c2), which show the outputs for these beams. Finally, Fig. 8(d1) shows a beam where both the $| R 〉$ , and $| L 〉$ polarizations have m = 0, and 1 components. Figure 8(d2) shows the sorter output for this beam. Once again, each eigenmode is fully separated and identifiable. These results demonstrate the breath of available state space in which spatially structured information can be decoded by the PBOE angular momentum sorter.

Fig. 8 (a1-d1) Structured intensity and polarization patterns of the hybrid modes composed of different superpositions of the $L G_{0}^{\pm 1}$ , $L G_{0}^{\pm 3}$ , $L G_{0}^{\pm 5}$ , $L G_{1}^{\pm 1}$ , $L G_{1}^{\pm 3}$ , and $L G_{1}^{\pm 5}$ cavity modes. (a2-d2) The sorter output for each of the coresponding hybrid modes. The exact cavity mode components of the modes for each SAM state are annotated on the figure. The l = $\pm$ 1 components contribute twice the power to the total field as each of the others. All intensity profiles are normalized to their own maximum.

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The sorting results presented in this section demonstrate that the proposed PBOE device can decompose the OAM and SAM eigenstates of a complex mixed mode optical beam. We additionally showed that this device can determine the integer index m of a Laguerre-Gaussian mode even though this parameter is unrelated to the angular momentum of the beam. These capabilities could enable the readout of information encoded in the topological phase and intensity spatial cross-section of a structured optical beam. Furthermore, the device could be used to determine the modal composition of the output of laser cavities and fiber optics by identifying the relative magnitude of each Laguerre-Gaussian eigenmode including both SAM states.

4. Conclusion

The ability to manipulate the Pancharatnam phase of optical waves opens possibilities to control light in new ways, on size scales previously unobtainable, and with near perfect diffraction efficiencies. The host of technologies that has recently emerged to implement PBOEs has the potential to revolutionize optics. In this work, we have proposed a PBOE angular momentum sorting device that could play an important role in future communications systems and scientific measurement techniques. We have demonstrated the functionality of this device to sort simultaneously the OAM and SAM eigenstates of an optical beam in a single measurement, and shown that this enables the ability to characterize the modal composition of structured optical beams without the need for numerical decomposition or interferometry.

We defined the two-dimensional spatially varying rotation angle of the slow axis in an anisotropic material or structure that implements the two PBOEs used to generate the azimuth to linear phase geometric optical transform that simultaneously sorts the OAM and SAM eigenstates of an optical beam. We took advantage of the circular polarization selective nature of PBOEs to enable full characterization of the angular momentum eigenstates of a beam as opposed to previous OAM only methods. We showed that this enables the ability to determine in a single measurement the eigenmode composition of structured beams resulting from hybrid mode combinations. Finally, we showed that since the azimuthal to linear phase geometric transformation is implemented with a ring-to-point coordinate mapping, the device can additionally determine the m parameter of a $L G_{m}^{l}$ Laguerre-Gaussian modes. Since these and similar beams can be generated and modulated with specialty optics, laser cavities, and fiber optics, these explorations demonstrate the importance that such a PBOE device could have for future angular momentum based optical communication systems.

Acknowledgments

This work was funded by the U.S. Army Natick Soldier Research Development and Engineering Center. The author thanks Dr. Francisco Aranda and Dr. Michael Ghebrebrhan for thoughtful discussion on this topic and critical review of the manuscript. Also Brian Kimball, Dr. Nelson Tabiryan, Dr. Sarik Narsisyan, and Dr. David Roberts for helpful discussions on PBOE devices in general. This work has been approved for public release. NSRDEC PAO# U15-312.

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Pancharatnam-Berry optical element sorter of full angular momentum eigenstate

Abstract

1. Introduction

2. Pancharatnam-Berry angular momentum sorter

3. Sorting angular momentum of laser cavity modes

4. Sorting higher order and hybrid, laser cavity modes

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (7)

Optics Express