Abstract

Based on the formal analogy between classical nonseparability and quantum entanglement, we present a multi-ary encoding protocol exploiting the nonseparability of orbital angular momentum (OAM) and polarization for a hybrid vector beam. Such an encoding can be realized in high-dimensional state space by transforming OAM of the vector beam under the assistance of polarization, which is called “high-dimensional” encoding. It is shown that N-ary encoding using N-dimensional non-separable basis can be obtained by manipulating N/2 different OAM modes, which is equivalent to encoding log2N bits of information. It is also shown that the decoding of vector beams can be realized with very low cross talk. Compared with the encoding protocol transforming OAM modes of scalar beams, our encoding scheme, based on classical nonseparability of vector beams, can encode much more information. This is of great benefit to the optical communication.

© 2016 Optical Society of America

1. Introduction

During the past decades, one of the primary interests of the optical communication is to achieve higher data transmission capacity. To realize this goal, considerable researches have been done using different physical properties. In classical optics communications, the multiplexing of multiple independent data channels can be used to improve the information capacity, such as wavelength-division multiplexing, polarization-division multiplexing, and space-division multiplexing [1–3]. In quantum regimes, dense coding or super-dense coding has been proposed and demonstrated experimentally by using the quantum entanglement [4–8]. In recent years, as one way to describe the space degree of freedom (DoF) of light, orbital angular momentum (OAM) modes have been used widely to improve tremendously the system capacity ranging from classical optics communications to quantum information [9–20]. The light carrying the OAM possesses an azimuthal phase term exp(ilφ), where l and φ denote topological charge (TC) and azimuthal angle, respectively [21]. Inherent orthogonality of OAM modes with different TCs has been exploited, which is a very attractive for the data transmission. However, the larger the value of the OAM, the larger the aperture required at the receiving optical system [22, 23]. It is only this that restricts the maximum value of l that may be used.

Most recently, there is a growing interest in information encoding by using vector beams. It has been demonstrated experimentally that vector beams cannot only be used for mode division multiplexing, they are also considered as a state space with which to encode information [24–27]. Light beams with spatially inhomogeneous states of the polarization refer to vector beams [28], which can be generated such as using q-plate [29], laser cavity [30], interferometric approach [31], dielectric metasurfaces [32], etc. A key characteristic of the fields of vector beams is the coupling (nonseparability) between the spatial mode and polarization. Owing to the similarities of the mathematic structure between the nonseparability and the quantum entanglement, the classical nonseparability (entanglement) has been widely investigated recently [33–44]. Utilizing classical nonseparability, Milione et al. [27] have pointed out that the 4-dimensional state space of light’s polarization and space DoFs can be accessed by only transforming the polarization. This means that 2 bits of information can be encoded by manipulating only the polarization DoF. However, the dimensions of state spaces cannot be further increased because the polarization DoF is restricted to two-dimensional Hilbert space.

In this work, we propose and demonstrate experimentally multi-ary encoding/decoding in high-dimensional state space formed by multiple possible states of hybrid vector beams, which is called “high-dimensional” encoding/decoding. Based on classical nonseparability, using a modified polarizing Sagnac Interferometer [45], such “high-dimensional” encoding can be realized by transforming the OAM modes of vector beams under the assistance of polarization. We have investigated performance of quaternary and hexadecimal coding/decoding using four and sixteen possible vector beams, respectively. Actually, there is no physical limit to achieve higher multi-ary coding with our proposed structure. This implies that such “high-dimensional” encoding can be realized by using N possible states of vector beams. Moreover, we find that the realization of quaternary (hexadecimal) coding just involves two (eight) different OAM modes, which means that N-dimensional non-separable basis can be obtained by manipulating N/2 different OAM modes of vector beams. This is equivalent to encoding log2N bits of information. Based on classical nonseparability, we can encode twice as much information as compared to using OAM modes of scalar beams. It is also shown that vector beams can be efficiently decoded with very low mode crosstalk using a traditional Mach- Zehnder interferometer and spatial light modulators (SLM).

2. Theoretical analysis

2.1 The nonseparability of hybrid vector beams

The cylindrically symmetric vector beams have been known for a long time [28]. Other families of vector beams have also been demonstrated such as hybrid vector beams [46, 47], the full Poincaré beams [48], and arbitrary vector beams [49]. Herein, we pay attention to the hybrid vector beam, which implies the superposition of two linearly polarized components with the same topological charge and opposite sign. According to [50], the state of the hybrid vector beam with a unit-amplitude is expressed in the form as

|Ψl+=cosθ|l|H+sinθ|l|V,
where the kets |l and |l are OAM modes with the lof per photon representing the spatial modes in the infinite dimensional space H. |H and |VH2 stand for horizontal and vertical polarization. The symbol denotes the tensor product. The relative weight of the fields are determined by θ. When θ0 and θ90, the state |Ψl+ of the form in Eq. (1) can no longer be factored into a direct product of two states with polarization and OAM modes. And the nonseparability between the space and polarization DoFs of light depends on the angleθ. It is worth noting that although we adopt the ket-bra notation that is associated with quantum mechanics, the description of our experiment requires no invoking of quantum mechanics and we adopt this notation to emphasize the linear algebraic nature of different degrees of freedom [42, 43].

In order to quantify the nonseparability of hybrid vector beams, the degree of polarization (space) coherence Dpol(DOAM) has been defined [40, 51]. Herein we take Dpol as an example. According to [40, 51], the degree of polarization coherence can be described in the following formula

Dpol=1S0p(S1p)2+(S2p)2+(S3p)2,
where Sip(i=0,1,2,3) are the Stokes parameters. Normalized Stokes parameters in the horizontal and vertical polarization bases can be measured and calculated [50, 52].
S0p=I(0,0)+I(90,0)=|H|Ψl+|2+|V|Ψl+|2,S1p=I(0,0)I(90,0)=|H|Ψl+|2|V|Ψl+|2,S2p=I(45,0)I(135,0)=2Re(H|Ψl+V|Ψl+*),S3p=I(45,90)I(135,90)=2Im(H|Ψl+V|Ψl+*),
where I(0,0), I(45,0), I(90,0) and I(135,0) are the intensities of the output beams through succeeding a rotated linear polariser oriented at 0, 45, 90, and 135, respectively. And I(45,90) and I(135,90) can be obtained when the vector beams pass through a combination of quarter-wave plane (QWP) and linear polarizer. Here 45 (135) denotes the orientation of a QWP with respect to the horizontal axis, and 90 is the the orientation of a linear polariser. According to Eqs. (1)-(3), the degree of polarization coherence Dpol can be calculated as follows:
Dpol=|cos2θsin2θ|.
Thus, the Dpol is expressed as a function of θ. Namely, the nonseparability between the polarization and space DoFs is determined by θ [40]. When θ=45, the nonseparability is the strongest, and the state of the maximally non-separable vector beam can be obtained; when θ=0 or θ=90, the state in Eq. (1) becomes separable state and can be written as a product of two states, i.e., |Ψl+=|l|H.

Another characteristic of the non-separable state is that the measuring result for the polarization (space) DoF depends on the projective measurement on the space (polarization) [37, 42]. Here, we consider an arbitrary polarized state expressed as

|Ω=cosγ|H+eiεsinγ|V,
where γ and ε, respectively, stand for the longitude and latitude angles of a point in the Poincaré sphere which is a simple and convenient geometric representation [52]. Then projecting the state |Ψl+ in Eq. (1) to |Ω in Eq. (5), the final state can be obtained
|Ψf=Ω|Ψl+=cosγcosθ|l+eiεsinγsinθ|l.
Because OAM modes |l and |l carry helical phase structures exp(ilφ) and exp(ilφ) respectively, the intensity distribution corresponding to Eq. (6) can be expressed as
I|cosγcosθeilφ+eiεsinγsinθeilφ|2=12[1+cos(2γ)cos(2θ)+sin(2γ)sin(2θ)cos(2lφε)].
One can see that when the initial state |Ψl+ is separable (θ=0or θ=90), the spatial profile in Eq. (7), ignoring intensity, is irrelevant to the projection state |Ω defined by γ and ε. Conversely, for a non-separable state, such as the maximally non-separable state (θ=45), the final state is pure superposition with the opposite topological charge l, and the intensity distribution depends on the selection of the projection polarization state.

2.2 “High-dimensional” encoding

The formal analogy between quantum entanglement and classical nonseparability inspires our research interests to encode information using the classical nonseparability. Dense coding was first proposed by Bennett et al. [4] and was realized experimentally by Mattle et al. [5]. The original quantum dense coding involves two particles A and B, which constitutes a joint maximally entangled Bell state being shared by Alice and Bob. For the particle B (one bit of information), Bob then performs unitary transformation, the initial entangled state can be transformed into each of four orthogonal Bell states carrying two bits of information. Theoretically, Alice finally reads the encoded information by determining the Bell state of the two-particle system. Therefore, the aim to communicate two bits of information can be achieved by transforming one bit.

To realize information encoding utilizing classical nonseparability, we replace two particles with two DoFs of a vector beam. Herein, the two DoFs are polarization and space that play the roles of particles A and B, respectively. Thus the non-separable state of the vector beam can be considered as analog of entangled Bell state between different DoFs. In fact, Milione et al. [24, 27] have shown that the information can be converted into symbols carried vector beams, and 2 bits of information can be encoded by applying the identity and three Pauli operators to the polarization DoF. Although the polarization is manipulated conveniently, the Hilbert space associated with the polarization is restricted to two dimensions. Thus, only 4-dimensional state space can be obtained by manipulating the polarization DoF. In contrast, OAM modes can span an infinite dimensional Hilbert space in principle. This characteristic suggests the use of OAM modes, alone or coupled with the spin, as a resource to encode information in higher dimensional states [14, 26]. To simplify our implementation, we consider a maximally non-separable state of the vector beam, Eq. (1) can be thus rewritten as

|Ψl+=12(|l|H+|l|V).
Figure 1(a) depicts the composition of a vector vortex beam corresponding to the state in Eq. (8) with l=1. The next step is to encode information by performing unitary transformation for the OAM modes in Eq. (8). For the initial state with the fixed OAM number l in Eq. (8), similar to the manipulation of the polarization, Bob firstly applies the identity (σ˜0) and three Pauli operators (σ˜x,σ˜y,σ˜z) to the OAM modes of a vector beam |Ψl+ [53]. Here, the Pauli operators with the tilde can be given by
σ˜0=|hh|+|vv|,σ˜x=|dd||aa|,σ˜y=|ll||ll|,σ˜z=|hh||vv|,
where |h (|v) refers to the OAM equivalent of the linear polarization bases |H (|V),|aand |d correspond to the angle antidiagonal and diagonal linear polarizations respectively, which can be defined as [53, 54]
|h=12(|l+|l),|v=1i2(|l|l),|a=eiπ42(|l+i|l),|d=eiπ42(|li|l).
Therefore, the results of transformations of the form in Eq. (8) can be given by
σ˜0|Ψl+=12(|l|H+|l|V)|Ψl+,σ˜x|Ψl+=i2(|l|H|l|V)|Φl,σ˜y|Ψl+=12(|l|H|l|V)|Ψl,σ˜z|Ψl+=12(|l|H+|l|V)|Φl+.
As can be seen, the initial non-separable state of the vector beam will be turned into one of the four states of vector beams that are orthogonal to each other. For the OAM number l=1, the transverse profiles of vector beams, corresponding to the four orthogonal states, are shown in Fig. 1(b).

 figure: Fig. 1

Fig. 1 Schematic showing the polarization distributions of vector beams. (a) OAM (l=1) mode and polarization of a vector beam Ψ1+ are non-separable. (b) Spatially varying polarizations of vector beams after applying the identity (σ˜0) and three Pauli operators (σ˜x,σ˜y,σ˜z) to the space degree of freedom of |Ψ1+. (c) High order hybrid vector beams (l=2) are generated by varying the OAM number based on (b).

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Furthermore, by virtue of the infinite dimensional Hilbert space associated with OAM modes, various kinds of hybrid vector beams can be generated by varying the order l of the OAM. In order to ensure the generated vector beams are orthogonal to each other, we vary the order l by exploiting a spin-controlled gate. That is |l|H|l+Δl|H and |l|V|lΔl|V where Δl denotes the gap between the OAM numbers l and l+Δl. Thus, Eq. (8) becomes

|Ψl+Δl+=12(|l+Δl|H+|lΔl|V).
Again Bob applies the identity (σ˜0) and three Pauli operators (σ˜x,σ˜y,σ˜z) to the space DoF of the state of the vector beam |Ψl+Δl+. Similarity, |Ψl+Δl+ can be transformed into one of the four states of vector beams |Ψl+Δl+,|Ψl+Δl,|Φl+Δl+ and |Φl+Δl. Figure 1(c) shows that four higher order hybrid vector beams can be obtained based on the vector beams in Fig. 1(b) when Δl=1. Therefore, by applying different operations to the space DoF of the state |Ψl+, the initial vector beam can be one of multiple states of vector beams. These possible states are orthogonal to each other and pose high-dimensional non-separable basis. This implies that N-dimensional encoding based on this non-separable basis can be realized by transforming N/2 different OAM modes.

3. Experiment implementation

In order to realize “high-dimensional” encoding/decoding, we consider proof of the principle experimental setup, as shown in Fig. 2. The schematic representation of the setup contains three parts: the preparation and measurement of the non-separable vector beam; Bob’s station for encoding the messages; finally, Alice’s analyzer to identify signals sent by Bob. The experimental setup for generating a hybrid vector beam is illustrated in Fig. 2(a). Light from a 632.8 nm helium-neon (He-Ne) laser passes thought a half-wave plate (HWP) and a polarizing beam splitter (PBS), which can be used to modulate the intensity. Subsequently, the beam propagates through an optical element group comprising of a HWP and a QWP. When the orientation of the fast axis of this QWP is set to 45 with respect to the horizontal axis, the state of beam is then transformed as

|0|H|0{|H+[sin(4θ1)icos(4θ1)]|V},
where |0|H (|0represents fundamental Gaussian mode) refers to the state of input beam, and θ1 represents the orientation of the fast axis of the HWP. Obviously, the relative phase between the horizontal and vertical polarization components can be controlled by rotating the HWP. Then the beam enters the trapezoid Sagnac interferometer where a vortex phase plate (VPP1) and a bunch of wave plates consisting of a PBS and two HWP are inserted. Since the two counter-propagating beams share the same optical path and elements [see Fig. 2(a)], the interferometer is automatically phase-stable. The TC l=+1 of VPP1 is chosen, with respect to the horizontally polarized beam. As a result, the VPP1 is able to transform the modes of input beams, |0|H|+1|H and |0|V|1|V. By setting the angle of one HWP to 22.5 in the interferometer, the state of the vector beam after the trapezoid Sagnac interferometer can be described by
|Ψ1+cos(θ)|+1|H+sin(θ)[sin(4θ1)icos(4θ1)]|1|V,
where θ2 represents the orientation of the fast axis of the other HWP. As can be seen, using the two optical element groups [see Fig. 2(a)], the relative phase and intensity of the final state can be controlled by rotating θ1 and θ respectively. Setting θ1=22.5, Eq. (14) becomes

 figure: Fig. 2

Fig. 2 Experimental setup for “high-dimensional” encoding using the classical nonseparability. (a) The preparation of the classical nonseparability of a vector beam. (b) The measurement of the nonseparability. (c) and (d) show Bob’s encoding and Alice’s decoding, respectively. Legend of the main components: HWP – half-wave plant; QWP – quarter-wave plant; BD – beam dumpers; PBS – polarizing beam splitter; BS – beam splitter; DP – dove prism; VPP – vortex phase plate; SLM – spatial light modulator; M – mirror; D – detector (camera or power meter).

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|Ψ1+cosθ|+1|H+sinθ|1|V.

To confirm the reliability and availability of our scheme, we perform measurements of the degree of the polarization coherence Dpol and intensity profiles of the non-separable beam. The measurements can be carried out by using six polarization bases (H,V,A,D,Land R for horizontal, vertical, antidiagonal, diagonal, left-, and right-hand circular, respectively) that are realized experimentally by utilizing a HWP, a QWP and a PBS [35], as shown in Fig. 2(b). According to Eqs. (2) and (3), the Dpol can be quantitatively assessed using the recorded intensity. Figure 3(a) displays the result for the Dpol as a function of θ. The red round dots and black solid lines represent the experimental measurements and the theoretical results, respectively. In addition, by setting θ=45 (Dpol=0), Fig. 3(b) shows the experimental measurements for intensity patterns at six polarization bases (H,V,A,D,Land R), which are recorded with a camera (CCD), and corresponding theoretical results are given in insets. It can be seen that in Fig. 3 the experimental results are in good agreement with the theoretical calculations, which verifies the classical nonseparability between different DoFs for such a beam.

 figure: Fig. 3

Fig. 3 The experimental measurement of the nonseparability of vector beams. (a) The measurement of degree of polarization coherence Dpol as a function of θ. Solid lines and round dots represent the theoretical and experimental results, respectively. (b) For the maximum coupling strength θ=45, the intensity distribution desponds on the polarization settings in front of the CCD where H,V,A,D,L and R represent horizontal, vertical, antidiagonal, diagonal, left-, and right-hand circular. Insets show theoretical intensity pattern.

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After generating the vector beam of the form in Eq. (15), Bob encodes information by transforming the OAM modes. In Fig. 2(c), information encoding can be achieved using a modified Sagnac interferometer with an additional mirror and a dove prism (DP2). In the interferometer, a VPP2 and a DP1 are inserted. Let us select the TC=+Δl of the VPP2 and define σ2 as the angle between the base of the DP1 and the plane of the interferometer, with respect to the horizontally polarized beam. Just as [55, 56], using the polarization Sagnac interferometer, we carry out a spin-controlled gate over OAM modes. The VPP2 and DP1 can change the OAM modes and the relative phase of the input vector beam, respectively. That is |H|+1eiσ|H|1Δl and |V|1ei(1+Δl)σ|V|+1+Δl. The output light from the interferometer then passes though a silver mirror. By setting the angles σ2 of the DP1 to 0 and π(4+2Δl) respectively, two states of vector beams can be obtained and expressed as

|Ψ1+|Ψ1+Δl±cosθ|+1+Δl|H±sinθ|1Δl|V.
Subsequently, if the vector beams pass thought the DP2@0, Eq. (16) becomes
|Ψ1+|Φ1+Δl±cosθ|1Δl|H±sinθ|+1+Δl|V,
Obviously, for each of fixed Δl, the state in Eq. (15) can become one of four states of vector beams. By varying Δl, we can also obtain various states of vector beams. When the initial state of the vector beam is maximally non-separable (θ=45), generated states arising from the state in Eq. (15) are orthogonal to each other and pose a high-dimensional non-separable basis, which can be used to encode multi-ary symbols. Table 1 clearly shows Bob’s coding operations. In our experiment the operation of the DP2@0 can be denoted by ‘1’ and ‘0’ where ‘1’ implies that the DP2 is inserted in the optical path and ‘0’ represents the operation without inserting DP2.

Tables Icon

Table 1. Overview of possible manipulations and results for the Bob’s encoding.

After encoding, the output beam from the Bob’s encoding station is then sent to Alice’s analyzer to perform decoding. Alice can obtain the encoded information by determining the state of vector beams. The process of decoding is shown in Fig. 2(d). A traditional Mach-Zehnder interferometer with a PBS and a BS is used [27]. To obtain constructive (destructive) interference at output A (B), a HWP with the fast axes at 45 and a DP3 @0 are inserted in two arms of the interferometer, respectively. The following forms provide the corresponding mapping between input and output beams at ports A and B:

|Ψ1+Δl+|1Δl|HportA,|Ψ1+Δl|1Δl|HportB,|Φ1+Δl+|1+Δl|HportA,|Φ1+Δl|1+Δl|HportB.
The whole states obtained by Alcie are divided into categories, which are independent on OAM numbers. |Ψ1+Δl+ and |Φ1+Δl+ vector beams will constructively (destructively) interfere at output A (B), and |Ψ1+Δl and |Φ1+Δl vector beams will destructively (constructively) interfere at output A (B). Next, Alice’s decoding can be achieved by identifying OAM modes at outputs A and B, respectively. About the identification of OAM modes, several methods have been put forward [9, 57–60]. Herein we use a hologram grating as a mode detector for a specific OAM mode [9, 57]. In the experiment, the beam from the output A (B) propagates through a SLM (HOLOEYE-LOTO) encoded a forked binary-grating hologram [57]. At the ±1 diffraction orders A1 and A2 (B1 and B2), the intensity profiles of the back-converted beams are then captured by two detectors which are placed in the focal plane of the lens. Finally we can obtain four spatially separated regions A1, A2, B1 and B2. According to Eq. (5) in [9], one can see that only one of the OAM modes (e.g. TC=l) can be converted back to a fundamental Gaussian beam with a bright spot when an inverse spiral phase mask (TC=l) is used. The others are still OAM beams, but with updated charge values. Then, based on the inverse spiral phase mask and the position of the bright spot, Alice’s decoding can be realized. In addition, to quantitatively assess this decoding, cross talk is calculated according to [25, 27].

We first use four possible vector beams of |Ψ1+, |Ψ1, |Φ1+ and |Φ1 (Δl=0) to code 4 quaternary base numbers 1, 2, 3 and 4, respectively, as described in Fig. 4(a). After restricting the space DoF to the two-dimensional Hilbert subspace, Bob can code information by applying the σ˜0,σ˜x,σ˜y and σ˜z operators to the space DoF of the |Ψ1+ vector beam respectively, as described in Table 1. The result of Alice’s decoding is given in Fig. 4(b). The on-axis intensity of each back-converted beam at four ports A1, A2, B1 and B2 is obtained by using a lens, a pin-hole and a power meter (not shown in Fig. 2). As can be seen, the four vector beams |Ψ1+, |Ψ1, |Φ1+ and |Φ1 correspond to four outputs respectively. To quantitatively assess this, mode crosstalk for each channel was measured. Herein normalized cross talk is shown in each of the four regions A1, A2, B1 and B2 of Fig. 4(b), which is calculated by recording the intensity in each one of the four outputs and then dividing that of all four outputs. Obviously, there is very low cross talk (<6.6%) for any channel, which is mostly attributed to the conversion efficiency of the SLM and misalignment of the interferometer. This means that the 4-dimensional state space of polarization and space of light can be obtained by transforming the OAM modes. And coding 2 bits of information is accomplished by operating two different OAM modes. Interestingly, it is noted that this is a result of the information encoding as an analogy of quantum dense coding.

 figure: Fig. 4

Fig. 4 (a) Bob’s coding for quaternary (1, 2, 3, 4) and hexadecimal (1, 2…7, 8…14, 15, 16) base numbers is shown and corresponds to 4 vector beams (|Ψ1+, |Ψ1, |Φ1+, |Φ1) and 16 vector beams (|Ψ1+, |Ψ1|Φ4+, |Φ4) respectively. (b) The normalized intensity of the center of the back-converted beam at four outputs is obtained with a power meter respectively when applying the identity (σ˜0) and three Pauli operators (σ˜x,σ˜y,σ˜z) to the space DoF of a |Ψ1+ vector beam. Normalized cross talk is shown in each output port.

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Furthermore, we have also prepared 16 states of vector beams (|Ψ1+, |Ψ1|Φ4+, |Φ4) representing hexadecimal base numbers 1, 2…7, 8…15, 16, respectively. At the transmitter side, each 16-base number in the symbol sequence corresponds to a vector beam, as shown in Fig. 4(a). The big square picture, consisting of 16 small pictures, can be considered as a hexadecimal symbol. When SLMs carry a forked binary-grating hologram with topological charge l=3 respectively, the partially decoding results are depicted in Fig. 5(a). Intensity patterns are recoded with a CCD at four output ports respectively when performing unitary transformation on the space DoF of a |Ψ1+ vector beam, as described in Table 1. In these patterns, each small picture at each output port corresponds to the diffraction pattern of the back-converted beam. One can see from the results that, for four vector beams with Δl=2 (|Ψ3+, |Ψ3, |Φ3+ and |Φ3), each back-converted beam forms a bright spot which is emerged in one of the four outputs. Other vector beams corresponds to ‘doughnut’ shape with no intensity at the center. Then Alice can thus decode the messages Bob sent according to the position in which the bright spots at the center of the beam are emerged. The obtained results shown in Fig. 5(a) are determined as hexadecimal base 9, 10, 11 and 12 which correspond to the outputs A1, B1, A2 and B2 respectively. To quantitatively assess this, mode crosstalk for each channel is measured, as shown in Fig. 5(b). The intensity of the center of the back-converted beams is recorded using a power meter behind a pin-hole. Normalized cross talk is calculated by measuring the intensity in each of outputs and then dividing that sum by sum of the sums of all sixteen regions which correspond to l=1,2,3,4. As can be seen, there is <13.6% mode crosstalk for any channel. Therefore, based on the 16 orthogonal states of vector beams, hexadecimal encoding/decoding can be realized, and this requires only 8 different OAM modes (±1,±2,±3and ±4). Although “high-dimensional” encoding discussed in this work is quaternary and hexadecimal coding, the results can be extended to higher multi-ary encoding in the high-dimensional state space.

 figure: Fig. 5

Fig. 5 (a) Experimentally measured transverse intensity profiles of decoded vector beams of |Ψ3+, |Ψ3, |Φ3+ and |Φ3 which correspond to hexadecimal base numbers 9, 10, 11 and 12 respectively. Intensity images recorded with a CCD at four outputs (A1, A2, B1 and B2) indicate the resulting transformation of vector beams after a forked binary-grating with l=3 encoded on the SLM, a lens and a pin-hole, respectively. Hexadecimal base number 9, 10, 11 and 12 can be decoded based on the position of the bright spots at the center of beam. (b) Normalized cross talk matrix for hexadecimal coding/decoding is shown.

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In addition, we have also taken quaternary coding/decoding for example to investigate the impact of the change of the nonseparability of vector beams on information coding/decoding results. After generating the state of the vector beam in Eq. (15), we characterize our encoding/decoding by varying the angle θ, measuring all output states in the coding region, and recording the on-axis intensity of each beam at four output ports A1, A2, B1 and B2, respectively. The results of these measurements are shown in Fig. 6(a). From these data, for each of initial vector beams (fixed angle θ), based on the position of the on-axis high-intensity, Alice can detect each message that Bob sent. One can see that the mode crosstalk gradually increases with the variation of the nonseparability between the space and polarization DoFs (θ<45or θ>45). To quantitatively assess this, taking θ=0, θ=15, θ=60 and θ=90 for examples, we measure mode crosstalk respectively. Normalized cross talk is defined as the method mentioned in Fig. 4(b), as shown in Figs. 6(b1)-6(b4). As can be seen, mode crosstalk depends on the nonseparability of vector beams. When θ=0 or θ=90, there is as high as 47.9% cross talk. This leads to that Alice only distinguishes the two different messages after applying the σ˜0,σ˜x,σ˜y and σ˜z operators to the space DoF. Compared with the coding/decoding mentioned in Fig. 4(b), only 1 bit of information can be encoded and decoded. This is due to the initial state without nonseparability. Thus we can encode twice as much information when using a non-separable state as compared to using a separable state.

 figure: Fig. 6

Fig. 6 (a) The decoding result for quaternary base numbers as a function of nonseparability referring to θ. The normalized on-axis intensity of each beam at four outputs (A1, A2, B1 and B2) is recoded when the nonseparability is changed ranging from the maximum (θ=45) to minimum (θ=0 or θ=90). (b) For quaternary coding/decoding, normalized mode crosstalk matrix is shown, which depends on the nonseparability of vector beams. The cross talk of (b1)-(b4) can be obtained when θ=0, θ=15, θ=60 and θ=90 respectively.

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The process of Alice’s decoding has been accomplished by observing the position of a bright spot at the center of the back-converted beam obtained by the inverse spiral phase mask encoded on SLMs, as described above. Although this decoding scheme has been used widely in the classical optics communication [9, 16], it cannot simultaneously display differentiation of the whole symbols encoded in vector beams. To circumvent this issue, we design a single composite hologram encoded on the SLM to displace the inverse spiral phase mask, as shown in Fig. 7(a), which is used to identify simultaneously differentiation of the input beams (symbols). This hologram consists of two phase patterns with l=1 and l=3 that create the horizontal and vertical diffraction orders, which gives an 3×3 array of diffraction patterns and can test 9 different OAM states simultaneously [14]. When a fundamental Gaussian beam illuminates this hologram, the sketch of the far-field diffraction patterns is obtained and shown in Fig. 7(b). Figure 7(c) shows a subset of results from hexadecimal base numbers 1, 2…7, 8…15, 16 using 16 vector beams [Fig. 4(a)]. Intensity images from two outputs A and B of the interferometer are recorded with a CCD array placed behind the SLMs respectively. Except for the center of the CCD images, other regions correspond to 16 vector beams respectively, which are determined as the hexadecimal base number. Figure 7(c) shows that the hexadecimal base numbers 8 (Φ2) and 9 (Ψ3+) are identified in the light of the position of the bright spot at the center of the diffraction beam. To quantitatively assess this, mode crosstalk is also calculated. According to [27], cross talk is defined here as the detection of light in a square region at the center of beams. For each input state, normalized mode crosstalk is calculated by summing the pixels in each of the sixteen square regions and then dividing that sum by sum of the sums of all sixteen square regions. Figure 7(d) shows normalized mode crosstalk for partial input states of vector beams. As can be seen, there is as high as 8.3% cross talk for these input states. Compared to Fig. 5(b), the difference of cross talk is most likely due to lack of spatial filtering. Even so, hexadecimal encoding/decoding can be still realized.

 figure: Fig. 7

Fig. 7 (a) Diagrams of a 3×3 vortex grating consisting of l=1 and l=3. (b) Schematic of far-field diffraction patterns and the OAM state distribution when this hologram is illuminated by a fundamental Gaussian beam. (c) The partially experimental results of decoding hexadecimal base numbers utilizing this composite hologram. The vector beams of Φ2+ (hexadecimal number 11) and Ψ3+ (hexadecimal number 13) can be determined based on the position of the bright spots at the beam center. (b) Normalized cross talk for hexadecimal coding/decoding is shown.

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It is noted that only 16 vector beams can be determined simultaneously utilizing the present composite hologram. In fact, integrating 5×5 binary Dammann vortex gratings have been designed to meet the demand of the practical application [61]. This grating can detect OAM states with the range from + 12 to −12. Therefore, in our work we can also encode 32-ary base number 1, 2, 3…31, 32 using 32 vector beams of Ψ1+,Ψ1...Φ8+ and Φ8, respectively. In fact, the key point of decoding process is the identification of OAM states according to Eq. (20). More generally, this decoding process can be also accomplished by transforming the output beams at Ports A and B using the wave front transformation from Cartesian to log-polar coordinate which is a more efficient and general method [59]. As pointed out, interferometric devices and holog ram can successfully work even at the single-photon level [62–64], which can be reached by attenuating the laser beam. Thus the present encoding/decoding scheme should be suitable for the case of the single photon in principle.

4. Conclusion

In conclusion, we have proposed and demonstrated experimentally multi-ary encoding schemes in the high-dimensional state space by exploiting the nonseparability of space and polarization DoF for a hybrid vector beam. Based on a modified polarizing Sagnac Interferometer, such “high-dimensional” encoding/decoding scheme can be completed by transforming OAM modes of the initial vector beam. As an example, quaternary and hexadecimal encoding/decoding can be successfully performed. The obtained results indicate that N-dimensional encoding/decoding can be realized using N-dimensional non-separable basis which is obtained by manipulating N/2 different OAM modes of vector beams. Compared with the encoding protocol using OAM-carrying scalar beams, our schemes can encode much more information. Additionally, with the same initial intensity profile, the vector vortex beam shows substantially lower scintillation than the scalar vortex beam in free-space optical communication [65]. This means that the atmospheric effects can be mitigated by using the vector beam carrying information. Therefore our scheme may have potential applications in many practice cases.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grants No. 11574031 and No.61421001.

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References

  • View by:

  1. Y. Han and G. Li, “Coherent optical communication using polarization multiple-input-multiple-output,” Opt. Express 13(19), 7527–7534 (2005).
    [Crossref] [PubMed]
  2. D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
    [Crossref]
  3. P. J. Winzer, “Making spatial multiplexing a reality,” Nat. Photonics 8(5), 345–348 (2014).
    [Crossref]
  4. C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69(20), 2881–2884 (1992).
    [Crossref] [PubMed]
  5. K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76(25), 4656–4659 (1996).
    [Crossref] [PubMed]
  6. C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, “Complete deterministic linear optics Bell state analysis,” Phys. Rev. Lett. 96(19), 190501 (2006).
    [Crossref] [PubMed]
  7. J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
    [Crossref]
  8. Y.-B. Sheng, F.-G. Deng, and G.-L. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82(3), 032318 (2010).
    [Crossref]
  9. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
    [Crossref]
  10. A. E. Willner, J. Wang, and H. Huang, “Applied physics. A different angle on light communications,” Science 337(6095), 655–656 (2012).
    [Crossref] [PubMed]
  11. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
    [Crossref] [PubMed]
  12. A. J. Willner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, “Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40(24), 5810–5813 (2015).
    [Crossref] [PubMed]
  13. H. Huang, G. Milione, M. P. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
    [Crossref] [PubMed]
  14. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
    [Crossref] [PubMed]
  15. D. Zhang, X. Feng, and Y. Huang, “Encoding and decoding of orbital angular momentum for wireless optical interconnects on chip,” Opt. Express 20(24), 26986–26995 (2012).
    [Crossref] [PubMed]
  16. J. Du and J. Wang, “High-dimensional structured light coding/decoding for free-space optical communications free of obstructions,” Opt. Lett. 40(21), 4827–4830 (2015).
    [Crossref] [PubMed]
  17. P. Li, Y. Sun, Z. Yang, X. Song, and X. Zhang, “Classical hypercorrelation and wave-optics analogy of quantum superdense coding,” Sci. Rep. 5, 18574 (2015).
    [Crossref] [PubMed]
  18. J. Leach, E. Bolduc, D. J. Gauthier, and R. W. Boyd, “Secure information capacity of photons entangled in many dimensions,” Phys. Rev. A 85(6), 060304 (2012).
    [Crossref]
  19. G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
    [Crossref] [PubMed]
  20. M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
    [Crossref]
  21. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref] [PubMed]
  22. R. L. Phillips and L. C. Andrews, “Spot size and divergence for Laguerre Gaussian beams of any order,” Appl. Opt. 22(5), 643–644 (1983).
    [Crossref] [PubMed]
  23. M. J. Padgett, F. M. Miatto, M. P. J. Lavery, A. Zeilinger, and R. W. Boyd, “Divergence of an orbital-angular-momentum-carrying beam upon propagation,” New J. Phys. 17(2), 023011 (2015).
    [Crossref]
  24. G. Milione, T. A. Nguyen, D. A. Nolan, E. Karimi, S. Slussarenko, L. Marrucci, and R. R. Alfano, “Superdense coding with vector vortex beams: a classical analogy of entanglement,” in Frontiers in Optics, OSA Technical Digest (online) (Optical Society of America, 2013), paper FM3F.4.
  25. G. Milione, M. P. J. Lavery, H. Huang, Y. Ren, G. Xie, T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, R. R. Alfano, and A. E. Willner, “4 × 20 Gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de)multiplexer,” Opt. Lett. 40(9), 1980–1983 (2015).
    [Crossref] [PubMed]
  26. Y. Zhao and J. Wang, “High-base vector beam encoding/decoding for visible-light communications,” Opt. Lett. 40(21), 4843–4846 (2015).
    [Crossref] [PubMed]
  27. G. Milione, T. A. Nguyen, J. Leach, D. A. Nolan, and R. R. Alfano, “Using the nonseparability of vector beams to encode information for optical communication,” Opt. Lett. 40(21), 4887–4890 (2015).
    [Crossref] [PubMed]
  28. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
    [Crossref]
  29. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
    [Crossref] [PubMed]
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2016 (2)

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10(5), 327–332 (2016).
[Crossref]

P. Li, B. Wang, X. Song, and X. Zhang, “Non-destructive identification of twisted light,” Opt. Lett. 41(7), 1574–1577 (2016).
[Crossref] [PubMed]

2015 (15)

X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2(7), 611–615 (2015).
[Crossref]

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92(2), 023833 (2015).
[Crossref]

S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magaña-Loaiza, and R. W. Boyd, “State transfer based on classical nonseparability,” Phys. Rev. A 92(2), 023827 (2015).
[Crossref]

E. Karimi and R. W. Boyd, “Classical entanglement?” Science 350(6265), 1172–1173 (2015).

Z.-C. Ren, L.-J. Kong, S.-M. Li, S.-X. Qian, Y. Li, C. Tu, and H.-T. Wang, “Generalized Poincaré sphere,” Opt. Express 23(20), 26586–26595 (2015).
[Crossref] [PubMed]

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015).
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M. J. Padgett, F. M. Miatto, M. P. J. Lavery, A. Zeilinger, and R. W. Boyd, “Divergence of an orbital-angular-momentum-carrying beam upon propagation,” New J. Phys. 17(2), 023011 (2015).
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G. Milione, M. P. J. Lavery, H. Huang, Y. Ren, G. Xie, T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, R. R. Alfano, and A. E. Willner, “4 × 20 Gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de)multiplexer,” Opt. Lett. 40(9), 1980–1983 (2015).
[Crossref] [PubMed]

Y. Zhao and J. Wang, “High-base vector beam encoding/decoding for visible-light communications,” Opt. Lett. 40(21), 4843–4846 (2015).
[Crossref] [PubMed]

G. Milione, T. A. Nguyen, J. Leach, D. A. Nolan, and R. R. Alfano, “Using the nonseparability of vector beams to encode information for optical communication,” Opt. Lett. 40(21), 4887–4890 (2015).
[Crossref] [PubMed]

H. Chen, Z. Chen, Q. Li, H. Lv, Q. Yu, and X. Yi, “Generation of vector beams based on dielectric metasurfaces,” J. Mod. Opt. 62(8), 638–643 (2015).
[Crossref]

A. J. Willner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, “Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40(24), 5810–5813 (2015).
[Crossref] [PubMed]

H. Huang, G. Milione, M. P. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
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J. Du and J. Wang, “High-dimensional structured light coding/decoding for free-space optical communications free of obstructions,” Opt. Lett. 40(21), 4827–4830 (2015).
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P. Li, Y. Sun, Z. Yang, X. Song, and X. Zhang, “Classical hypercorrelation and wave-optics analogy of quantum superdense coding,” Sci. Rep. 5, 18574 (2015).
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2014 (5)

P. J. Winzer, “Making spatial multiplexing a reality,” Nat. Photonics 8(5), 345–348 (2014).
[Crossref]

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113(6), 060503 (2014).
[Crossref] [PubMed]

R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Quantum entanglement of complex photon polarization patterns in vector beams,” Phys. Rev. A 89(6), 060301 (2014).
[Crossref]

O. S. Magaña-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of Angular Rotations Using Weak Measurements,” Phys. Rev. Lett. 112(20), 200401 (2014).
[Crossref] [PubMed]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39(8), 2411–2414 (2014).
[Crossref] [PubMed]

2013 (2)

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013).
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2012 (6)

D. Zhang, X. Feng, and Y. Huang, “Encoding and decoding of orbital angular momentum for wireless optical interconnects on chip,” Opt. Express 20(24), 26986–26995 (2012).
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J. Leach, E. Bolduc, D. J. Gauthier, and R. W. Boyd, “Secure information capacity of photons entangled in many dimensions,” Phys. Rev. A 85(6), 060304 (2012).
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J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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A. E. Willner, J. Wang, and H. Huang, “Applied physics. A different angle on light communications,” Science 337(6095), 655–656 (2012).
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S. Liu, P. Li, T. Peng, and J. Zhao, “Generation of arbitrary spatially variant polarization beams with a trapezoid Sagnac interferometer,” Opt. Express 20(19), 21715–21721 (2012).
[Crossref] [PubMed]

K. H. Kagalwala, G. D. Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7(1), 72–78 (2012).
[Crossref]

2011 (2)

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

A. Z. Khoury and P. Milman, “Quantum teleportation in the spin-orbit variables of photon pairs,” Phys. Rev. A 83(6), 060301 (2011).

2010 (12)

E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82(2), 022115 (2010).
[Crossref]

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization States,” Phys. Rev. Lett. 105(3), 030407 (2010).
[Crossref] [PubMed]

Y.-B. Sheng, F.-G. Deng, and G.-L. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82(3), 032318 (2010).
[Crossref]

M. A. Goldin, D. Francisco, and S. Ledesma, “Simulating Bell inequality violations with classical optics encoded qubits,” J. Opt. Soc. Am. B 27(4), 779–786 (2010).
[Crossref]

E. Nagali, L. Sansoni, L. Marrucci, E. Santamato, and F. Sciarrino, “Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding,” Phys. Rev. A 81(5), 052317 (2010).
[Crossref]

S. Slussarenko, V. D’Ambrosio, B. Piccirillo, L. Marrucci, and E. Santamato, “The polarizing Sagnac interferometer: a tool for light orbital angular momentum sorting and spin-orbit photon processing,” Opt. Express 18(26), 27205–27216 (2010).
[Crossref] [PubMed]

G. Milione, H. I. Sztul, and R. R. Alfano, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2010).
[Crossref]

X.-L. Wang, Y. Li, J. Chen, C.-S. Guo, J. Ding, and H.-T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18(10), 10786–10795 (2010).
[Crossref] [PubMed]

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18(10), 10777–10785 (2010).
[Crossref] [PubMed]

N. Zhang, X. C. Yuan, and R. E. Burge, “Extending the detection range of optical vortices by Dammann vortex gratings,” Opt. Lett. 35(20), 3495–3497 (2010).
[Crossref] [PubMed]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105(15), 153601 (2010).
[Crossref] [PubMed]

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82(3), 033833 (2010).
[Crossref]

2009 (2)

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009).
[Crossref] [PubMed]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

2008 (2)

J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4(4), 282–286 (2008).
[Crossref]

G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101(10), 100801 (2008).
[Crossref] [PubMed]

2007 (1)

2006 (2)

C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, “Complete deterministic linear optics Bell state analysis,” Phys. Rev. Lett. 96(19), 190501 (2006).
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L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

2005 (1)

2004 (1)

2003 (2)

Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, “Violation of a Bell-like inequality in single-neutron interferometry,” Nature 425(6953), 45–48 (2003).
[Crossref] [PubMed]

H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A 68(1), 012323 (2003).
[Crossref]

2002 (1)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[Crossref] [PubMed]

1999 (1)

1998 (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28(3), 361–374 (1998).
[Crossref]

1996 (1)

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76(25), 4656–4659 (1996).
[Crossref] [PubMed]

1992 (2)

C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. 69(20), 2881–2884 (1992).
[Crossref] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1983 (1)

Abouraddy, A. F.

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherency matrix tomography,” Opt. Lett. 39(8), 2411–2414 (2014).
[Crossref] [PubMed]

K. H. Kagalwala, G. D. Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7(1), 72–78 (2012).
[Crossref]

Ahmed, N.

A. J. Willner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, “Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40(24), 5810–5813 (2015).
[Crossref] [PubMed]

H. Huang, G. Milione, M. P. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Alfano, R. R.

H. Huang, G. Milione, M. P. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

G. Milione, M. P. J. Lavery, H. Huang, Y. Ren, G. Xie, T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, R. R. Alfano, and A. E. Willner, “4 × 20 Gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de)multiplexer,” Opt. Lett. 40(9), 1980–1983 (2015).
[Crossref] [PubMed]

G. Milione, T. A. Nguyen, J. Leach, D. A. Nolan, and R. R. Alfano, “Using the nonseparability of vector beams to encode information for optical communication,” Opt. Lett. 40(21), 4887–4890 (2015).
[Crossref] [PubMed]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

G. Milione, H. I. Sztul, and R. R. Alfano, “Stokes polarimetry of a hybrid vector beam from a spun elliptical core optical fiber,” Proc. SPIE 7613, 761305 (2010).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Alonso, M. A.

An Nguyen, T.

H. Huang, G. Milione, M. P. Lavery, G. Xie, Y. Ren, Y. Cao, N. Ahmed, T. An Nguyen, D. A. Nolan, M. J. Li, M. Tur, R. R. Alfano, and A. E. Willner, “Mode division multiplexing using an orbital angular momentum mode sorter and MIMO-DSP over a graded-index few-mode optical fibre,” Sci. Rep. 5, 14931 (2015).
[Crossref] [PubMed]

Andrews, L. C.

Badurek, G.

Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, “Violation of a Bell-like inequality in single-neutron interferometry,” Nature 425(6953), 45–48 (2003).
[Crossref] [PubMed]

Barnett, S.

Barnett, S. M.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref] [PubMed]

Baron, M.

Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, “Violation of a Bell-like inequality in single-neutron interferometry,” Nature 425(6953), 45–48 (2003).
[Crossref] [PubMed]

Barreiro, J. T.

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Figures (7)

Fig. 1
Fig. 1 Schematic showing the polarization distributions of vector beams. (a) OAM ( l=1 ) mode and polarization of a vector beam Ψ 1 + are non-separable. (b) Spatially varying polarizations of vector beams after applying the identity ( σ ˜ 0 ) and three Pauli operators ( σ ˜ x , σ ˜ y , σ ˜ z ) to the space degree of freedom of | Ψ 1 + . (c) High order hybrid vector beams ( l=2 ) are generated by varying the OAM number based on (b).
Fig. 2
Fig. 2 Experimental setup for “high-dimensional” encoding using the classical nonseparability. (a) The preparation of the classical nonseparability of a vector beam. (b) The measurement of the nonseparability. (c) and (d) show Bob’s encoding and Alice’s decoding, respectively. Legend of the main components: HWP – half-wave plant; QWP – quarter-wave plant; BD – beam dumpers; PBS – polarizing beam splitter; BS – beam splitter; DP – dove prism; VPP – vortex phase plate; SLM – spatial light modulator; M – mirror; D – detector (camera or power meter).
Fig. 3
Fig. 3 The experimental measurement of the nonseparability of vector beams. (a) The measurement of degree of polarization coherence D pol as a function of θ. Solid lines and round dots represent the theoretical and experimental results, respectively. (b) For the maximum coupling strength θ= 45 , the intensity distribution desponds on the polarization settings in front of the CCD where H,V,A,D,L and R represent horizontal, vertical, antidiagonal, diagonal, left-, and right-hand circular. Insets show theoretical intensity pattern.
Fig. 4
Fig. 4 (a) Bob’s coding for quaternary (1, 2, 3, 4) and hexadecimal (1, 2…7, 8…14, 15, 16) base numbers is shown and corresponds to 4 vector beams ( | Ψ 1 + , | Ψ 1 , | Φ 1 + , | Φ 1 ) and 16 vector beams ( | Ψ 1 + , | Ψ 1 | Φ 4 + , | Φ 4 ) respectively. (b) The normalized intensity of the center of the back-converted beam at four outputs is obtained with a power meter respectively when applying the identity ( σ ˜ 0 ) and three Pauli operators ( σ ˜ x , σ ˜ y , σ ˜ z ) to the space DoF of a | Ψ 1 + vector beam. Normalized cross talk is shown in each output port.
Fig. 5
Fig. 5 (a) Experimentally measured transverse intensity profiles of decoded vector beams of | Ψ 3 + , | Ψ 3 , | Φ 3 + and | Φ 3 which correspond to hexadecimal base numbers 9, 10, 11 and 12 respectively. Intensity images recorded with a CCD at four outputs (A1, A2, B1 and B2) indicate the resulting transformation of vector beams after a forked binary-grating with l =3 encoded on the SLM, a lens and a pin-hole, respectively. Hexadecimal base number 9, 10, 11 and 12 can be decoded based on the position of the bright spots at the center of beam. (b) Normalized cross talk matrix for hexadecimal coding/decoding is shown.
Fig. 6
Fig. 6 (a) The decoding result for quaternary base numbers as a function of nonseparability referring to θ. The normalized on-axis intensity of each beam at four outputs (A1, A2, B1 and B2) is recoded when the nonseparability is changed ranging from the maximum ( θ= 45 ) to minimum ( θ= 0 or θ= 90 ). (b) For quaternary coding/decoding, normalized mode crosstalk matrix is shown, which depends on the nonseparability of vector beams. The cross talk of (b1)-(b4) can be obtained when θ= 0 , θ= 15 , θ= 60 and θ= 90 respectively.
Fig. 7
Fig. 7 (a) Diagrams of a 3×3 vortex grating consisting of l=1 and l=3 . (b) Schematic of far-field diffraction patterns and the OAM state distribution when this hologram is illuminated by a fundamental Gaussian beam. (c) The partially experimental results of decoding hexadecimal base numbers utilizing this composite hologram. The vector beams of Φ 2 + (hexadecimal number 11) and Ψ 3 + (hexadecimal number 13) can be determined based on the position of the bright spots at the beam center. (b) Normalized cross talk for hexadecimal coding/decoding is shown.

Tables (1)

Tables Icon

Table 1 Overview of possible manipulations and results for the Bob’s encoding.

Equations (18)

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| Ψ l + =cosθ|l|H+sinθ| l |V,
D pol = 1 S 0 p ( S 1 p ) 2 + ( S 2 p ) 2 + ( S 3 p ) 2 ,
S 0 p =I( 0 , 0 )+I( 90 , 0 )= | H| Ψ l + | 2 + | V| Ψ l + | 2 , S 1 p =I( 0 , 0 )I( 90 , 0 )= | H| Ψ l + | 2 | V| Ψ l + | 2 , S 2 p =I( 45 , 0 )I( 135 , 0 )=2Re( H| Ψ l + V| Ψ l + * ), S 3 p =I( 45 , 90 )I( 135 , 90 )=2Im( H| Ψ l + V| Ψ l + * ),
D pol =| cos 2 θ sin 2 θ |.
|Ω=cosγ|H+ e iε sinγ|V,
| Ψ f =Ω| Ψ l + =cosγcosθ|l+ e iε sinγsinθ| l .
I | cosγcosθ e ilφ + e iε sinγsinθ e ilφ | 2 = 1 2 [ 1+cos( 2γ )cos( 2θ )+sin( 2γ )sin( 2θ )cos( 2lφε ) ].
| Ψ l + = 1 2 ( |l|H+| l |V ).
σ ˜ 0 =|hh|+|vv|, σ ˜ x =|dd||aa|, σ ˜ y =|ll|| l l |, σ ˜ z =|hh||vv|,
|h= 1 2 ( |l+| l ),|v= 1 i 2 ( |l| l ), |a= e i π 4 2 ( |l+i| l ),|d= e i π 4 2 ( |li| l ).
σ ˜ 0 | Ψ l + = 1 2 ( |l|H+| l |V )| Ψ l + , σ ˜ x | Ψ l + = i 2 ( | l |H|l|V )| Φ l , σ ˜ y | Ψ l + = 1 2 ( |l|H| l |V )| Ψ l , σ ˜ z | Ψ l + = 1 2 ( | l |H+|l|V )| Φ l + .
| Ψ l+Δl + = 1 2 ( | l+Δl |H+| lΔl |V ).
|0|H|0{ |H+[ sin( 4 θ 1 )icos( 4 θ 1 ) ]|V },
| Ψ 1 + cos( θ )| +1 |H+sin( θ )[ sin( 4 θ 1 )icos( 4 θ 1 ) ]| 1 |V,
| Ψ 1 + cos θ | + 1 | H + sin θ | 1 | V .
| Ψ 1 + | Ψ 1+Δl ± cosθ| +1+Δl |H±sinθ| 1Δl |V.
| Ψ 1 + | Φ 1+Δl ± cosθ| 1Δl |H±sinθ| +1+Δl |V,
| Ψ 1+Δl + | 1Δl |Hport A, | Ψ 1+Δl | 1Δl |Hport B, | Φ 1+Δl + | 1+Δl |Hport A, | Φ 1+Δl | 1+Δl |Hport B.

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