Abstract

Using orbital angular momentum (OAM) conservation in second-order nonlinear interaction processes, we create a classical to quantum optical network link in the OAM degrees of freedom of light via sum frequency generation, followed by spontaneous parametric down-conversion. Coherent OAM-carrying beams at 1550nm are up-converted to 525.5-nm OAM-carrying beams in the first crystal, and are used to pump a second crystal to generate non-degenerate OAM entangled photon pairs at 795nm and 1550nm. By switching the OAM carried by the classical part, OAM correlation in the quantum part is shifted. High-level OAM entanglements in two-dimensional subspaces are verified.

© 2015 Optical Society of America

1. Introduction

The field of optical communications consists of two quite different schemes, classical optical communications and quantum optical communications. These schemes differ in terms of the principles of their physical realization, such as the methods used for encoding, decoding and detection [1]. Information transmission capacity and security are the two most important indexes used to estimate the quality of an optical communication network. To enhance the transmission capacity of a single communication channel, multiplexing of the different degrees of freedom of light is used. In the early stages, time multiplexing, wavelength multiplexing and polarization multiplexing were used to dramatically increase communication speeds. More recently, multiplexing of the spatial degrees of freedom of light has been demonstrated in both free space [2, 3] and fiber-based [4] optical communications for orbital angular momentum (OAM)-carrying light beams, and the transmission rates have reached the Tbit/s level. Almost since the birth of quantum mechanics,quantum optical communications have been under developmentin pursuit of unconditional security in optical communications. Based on the basic principles of quantum mechanics, quantum communication offers unconditional security in principle. Encoding information in the OAM degrees of freedom of photons can enhance both the information-carrying capacity and noise tolerance in quantum key distribution [5–7].

Since it was first introduced by Allen et al. in [8], OAM-carrying light has been widely studied in many fields, including optical trapping and manipulation [9, 10], metrology [11–13], testing of the basic principles of quantum mechanics [14–17] and optical communications [2–4, 7]. The total OAM of a light beam is conserved in nonlinear optical interaction processes (e.g., second order [18–25], third order [26–28] and higher orders [29]). The conservation of OAM in nonlinear interaction processes is very useful for frequency conversion of OAM-carrying beams [18–25] and creation of OAM entangled photon pairs in two-dimensional subspaces [30–34] or in higher dimensions [16, 35]. Because of this conservation law, it is possible to create a link between a classical optical network and a quantum optical network encoded in terms of the OAM degrees of freedom of light. This link acts as a compatible interface to join a classical network and a quantum network to a hybrid network, which will then enable information transfer between two completely different communication systems.

In this article, we report the construction of a hybrid optical network by linking a classical network with a quantum network using the OAM degrees of freedom of light beams based on a process of sum frequency generation (SFG) followed by spontaneous parametric down-conversion (SPDC) in nonlinear crystals. A coherent OAM-carrying beam at the telecommunications wavelength of 1550nm is efficiently up-converted into a 525.5-nm OAM-carrying beam in the first periodically poled KTiOPO4 crystal (PPKTP1); then, the up-converted OAM-carrying beam is used to pump crystal PPKTP2 to generate non-degenerate OAM entangled photon pairs at 795nm and 1550nm. Because the total OAM is conserved in this cascade process, the OAM correlation of the quantum part is shifted by switching the OAM carried by the classical part. The dependence of the quantum OAM correlation on the OAM carried by the classical beam implies that information is transferred from the classical part to the quantum part in the OAM degrees of freedom of the light beams. In addition, high levels of OAM entanglements in the two-dimensional subspaces are verified via various methods, including interference, CHSH (Clauser-Horne-Shimony-Holt) inequality and quantum state tomography methods.

2. Experimental setup

The experiment consists of two parts, where the first part involves efficient up-conversion of an OAM-carrying light beam based on SFG in an external cavity, and the second part is SPDC using the up-converted OAM beams to generate non-degenerate OAM entangled photon pairs (see the simplified diagram at the top of Fig. 1). High-efficiency frequency conversion of OAM-carrying light beams in periodically poled nonlinear crystals has previously been studied in detail by the authorsin [19–22]. In the experiment presented here (Fig. 1), the external cavity is the same as that used in [22], and the cavity design parameters are described in detail in that paper. The type-I SFG and SPDC crystals were manufactured by Raicol Crystals, and all crystals have dimensions of 1 mm × 2mm × 10mm. The poling period of each crystal is 9.375µm, both end faces of each crystal are anti-reflection coated for wavelengths of 525.5 nm, 795nm and 1560nm, and the measured quasi-phasing matching temperatures of PPKTP1 and PPKTP2 are 39.4°C and 43.0°C,respectively. The power of the 1550-nm beam is 12mW (Toptica Prodesign laser), and that of the 795-nm beam is 700mW (Ti:sapphire laser, MBR110). The powers of the up-converted 525.5-nm beam for OAM indexes of 0 and −1 are 0.7mW and 0.15mW,respectively, after removing the two pump beams using filter F1(Thorlabs, FBH520-40,FESH1000). The up-converted OAM-carrying beam is then imaged into another crystal, PPKTP2, using a 150-mm lens (L1) for SPDC with a magnification of about 2, and thus the beam diameter at the center of PPKTP2 is approximately 130 µm. The PPKTP2 crystal emits non-degenerate photon pairs at 795nm and 1550nm. These two photons are separated by a dichromatic mirror, and they are then imaged separately to two spatial light modulators, A and B (SLMA/SLMB, PLUTO; active area of 15.36 × 8.64 mm2, pixel pitch size of 8 μm, and total number of pixels of 1920 × 1080) using 150-mm lenses (L2, L3) with magnifications of 15. The beam diameters at SLMA and SLMB are approximately 2mm. After transformation by SLMA and SLMB, the photons are then collected into single mode fibers (SMF1, SMF2) using 4-f imaging systems with similar lens groups (L4/L5 and L6/L7). The collected photons are detected using a Si avalanche detector and a free running InGaAs avalanche detector (IDQ 220), and the output signals of these detectors are sent to a coincidence device (Timeharp 260, Pico, with a coincidence window of 3.2ns).

 

Fig. 1 Experimental setup. The simplified upper diagram shows the information transfer direction. M1–M9: mirrors; VPP: vortex phase plate; F1–F3: filters; DM: dichromatic mirror; PPKTP1, 2: periodically poled KTP crystals; SMF1, 2: single mode fibers; L1–L3:150-mm lenses; L4, L7:200-mm lenses; L5, L6:100-mm lenses; PZT: piezoelectric transducer; SLMA, SLMB: spatial light modulators.

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3. Experimental results

The generated non-degenerate two-photon state that is correlated in the OAM can be expressed as [36]:

|Φ=ms=ms=+cms|ms,λs|lpms,λi.

Here, s and i stand for the signal and the idler respectively; |ms,λs and |lpms,λi denote the OAM eigenmodes of the signal and the idler, respectively; the OAMs carried by the signal (ms) and the idler (lpms) sum to give the OAM carried by the pump beamlp; |cms|2 is the probability of generationof a photon pair with OAM ofms, lpms; in the special case where lp=0, we can infer that cms=cms from the symmetry of the SPDC process.

To show that the total OAM is conserved in the cascade processes of SFG and SPDC and that the OAM information carried by the classical coherent light beams at 1550nm is transferred to the signal and idler photons at 1550nmand 795nm, respectively,we perform OAM correlation measurements for different signal and idler OAMs. Experimental results for OAM numbers of 0 and −1 carried by the 1550-nm coherent beam are shown in Fig. 2. Figure 2(a) shows a three-dimensional bar chart for the signal and idler OAM ranges from −5 to 5 for a pump beam OAM of 0. Because of misalignment, the coincidences of the nearby OAMs are larger than those of the non-adjacent OAMs. The maximum coincidence counts in 10 s are 31475 for the signal and idler OAMs of 0, and the smallest off-diagonal coincidence counts are approximately 100. Figure 2(b) shows that the corresponding coincidence counts for the signal and idler OAM sum to 0. We can infer that the spiral bandwidth of this two-color source is approximately 5.To increase the spiral bandwidth, we can use a larger beam waist, a shorter crystal length [37, 38] or change the phase-matching conditions by tuning the crystal temperature [39]. Figures 2(b) and 2(c) show the results for the pump OAM of −1, where the coincidence counts when the signal and idler OAMs sum to −1 are much higher than the other uncorrelated cases. The inset images in Figs. 2(a) and 2(c) show the intensity distributions of the 525.5-nm SFG beams. The slight spatial dependence of the conversion efficiency will lead to mode impurity in the SFG beam, and this mode impurity will lead to crosstalk from other OAM channels;however, the mode impurity in the present experiments is very small, and this crosstalk can simply be ignored. All these results are in good agreement with Eq. (1), which implies that the total OAM is conserved in the cascade processes, and that the OAM information flows from the classical part to the quantum part.

 

Fig. 2 Experimental results for measurementof the orbital angular momentum(OAM) correlation between the signal and idler photons for different pump OAMs at 1550nm. (a) and (c) show the coincidence counts in 10 s for signal and idler OAMs ranging from −5 to 5 for pump OAMs of 0 and−1, respectively; (b) and (d) show the coincidence counts when the signal and idler OAMs sum to give pump OAMs of 0 and −1, respectively.

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Equation (1) also implies that the signal and idler OAMs are entangled in a high-dimensional Hilbert space. To investigate the entanglement of the signal and idler photons, we simplify the measurement conditions to two-dimensional subspaces. For OAM entanglement in two-dimensional subspaces, we can use a Blochsphere to describe the state in a manner that is analogous to the polarization states. An equal superposition of |l and |l with arbitrary phase is represented by a point along the equator of the sphere [40–42]. In a fashion that is analogous to the polarization states, we define the superposition statesas follows:

|θ=12(eilθ|l+eilθ|l).

These states are represented by points along the equator of the sphere. The states |θ are also called sector states, and have 2l sectors of alternating phases.

To quantify the entanglement, we must demonstrate that the correlations between the signal and the idler are persistent for the superposition states. To verify this condition, we detect the photons in the sector states that were defined in Eq. (2) when oriented at different angles θA and θB. Based on Eqs. (1) and (2), the coincidence rate for detection of one photon in |θA and the other in |θB is given by

C(θA,θB)=|θA|θB||Φ|2cos2[l(θAθB)].
The high-visibility fringes of the joint probability are the signature of a two-dimensional entanglement. Any modal impurities will degrade the entanglementquality, which will in turn reduce the visibility of the fringes. By post-selection of the state in Eq. (1) into the two-dimensional subspaces, the states can be approximated as [32]:

|Φl=12(|l,795|l,1550+|l,795|l,1550).

The above states are normalized with respect to the chosen subspaces and are entangled in the OAM. By changing the phase masks on the SLMs, entangled states can be prepared with any given l. The experimental results in the two-dimensional subspace given by l=±1,±2 are shown in Figs. 3(a) and 3(b). By fixing the orientation of the hologram in SLMB at angle θB=0,π/4l, we measure the coincidence count in 10 s by changing the angle from 0 to π/l. The angle period is π/l and is dependent on l. The visibilities of the sinusoidal fringes are 89.36% and 84.11%, and 96.89% and 97.32% for θB=0,π/4l (l = 1, 2), respectively. All the visibilities are higher than 71%, which is sufficient to violate the CHSH inequality and imply the presence of the two-dimensional entanglement. Therefore,we further characterize the entanglement by measuring the CHSH inequality S parameter [43, 44]. The definition of S in our experiment for the angles θA and θB for the phase masks on the two SLMs is

 

Fig. 3 Experimental results for characterizationof the entanglement in two-dimensional subspaces.(a) and (b) show the coincidence counts of the signal and the idler as a function of phase mask angle θA when the phase mask angle θB=0,π/4l (l = 1, 2); (c) and (d) show the real and imaginary parts of the reconstructed density matrix for |Φ2.

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S=E(θA,θB)E(θA,θB)+E(θA,θB)+E(θA,θB).

The inequality is violated for values of |S| that are greater than 2. E(θA,θB) is calculated from the coincidence counts at specific sets of the hologram on the two SLMs:

E(θA,θB)=C(θA,θB)+C(θA+π2l,θB+π2l)C(θA+π2l,θB)C(θA,θB+π2l)C(θA,θB)+C(θA+π2l,θB+π2l)+C(θA+π2l,θB)+C(θA,θB+π2l).

For the entanglement state |Φl, the maximal violationof the inequality occurs,e.g., when θA=0,θB=π/8l,θA=π/4l and θB=3π/8l. The measured S parameters for |Φ1 and |Φ2 are 2.28 ± 0.016 and 2.82 ± 0.0021 with 17 and 390 standard deviations, respectively.

To describe an entangled statecompletely, reconstruction of the density matrix is necessary. To reconstruct the density matrix of |Φl, we follow quantum tomography methods that are analogous to the two-bit polarization entanglement states [45]. We define |L=|l and |R=|l. At least 16 projection measurements on the projection basis |L,|R,1/2(|L+|R) and 1/2(|Li|R) are required. As an example, the results for the state |Φ2 are shown in Figs. 3(c) and 3(d). The real and imaginary parts of the density matrix that were reconstructed using the maximum-likelihood method are shown in Figs. 3(c) and 3(d). The fidelity F=Φ|2ρe|Φ2 of the experimentally reconstructed density matrix with respect to the theoretical density matrix is 0.84 ± 0.0062. This discrepancy in the fidelity is caused by slightly different coupling efficiencies between the states |L and |R and the states 1/2(|L+|R) and 1/2(|Li|R). We use an OAM index of 2 as an example to illustrate the holograms that were used for each measurement in the experiments. The holograms that were used are listed in Fig. 4. Figures 4(a) and 4(b) show the holograms for the coincidence measurements shown in Fig. 1. Figures 4(c) and 4(d) show the definitions of the angles θA and θB, which are used in the measurements shown in Figs. 3(a) and 3(b). Figures 4(e)–4(h) are four orthogonal projection bases for the quantum tomography measurements shown in Figs. 3(c) and 3(d).

 

Fig. 4 Holograms used in the experimental measurements. (a) and (b) are the holograms for the single OAM states that were used in the measurements of Fig. 1; (c) and (d) show the definitions of θA and θB; (e)–(h) show holograms that were used for quantum tomography measurements.

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4. Summary and discussion

We have demonstrated proof of principle experiments for construction of a link between a classical telecommunications network and a quantum network usingthe OAM degrees of freedom of light based on cascade second-order nonlinear frequency conversion. An OAM-carrying beam at the telecommunications wavelength of 1550nm is up-converted into an OAM-carrying beam at 525.5nm using cavity-enhanced SFG, and then the up-converted OAM-carrying beam is used to pump a second crystal that is used for SDPC to generate non-degenerate two-color OAM entangled signal and idler photons at 795nm and 1550nm, respectively. The dependence of the OAM correlation on the classical beam is observed at 1550nm; when the OAM carried by the classical beam is switched, then the OAM correlation between the signal and the idler is also changed because the total OAM is conserved in the cascade processes. Therefore, the OAM information carried by the classical beam is transferred to the signal and idler photons. We also demonstrate high-level entanglement of the signal and idler photons in the two-dimensional OAM subspace. This two-color OAM entangled source is highly promisingfor quantum communications, because one of the photon wavelengths corresponds to the atom quantum memory of Rb85(where the bandwidth can be narrowed using a cavity), while the other wavelength is in a telecommunications band that is suitable for long distance transmission. Our work here will be helpful in fabrication of a compatible interface between a classical optical communication network and a quantum communication network based on OAM degrees of freedom; this hybrid network will offer both high capacity and superior security.

Acknowledgments

This work was supported by the National Fundamental Research Program of China (Grant No. 2011CBA00200) and the National Natural Science Foundation of China (Grant Nos. 11174271, 61275115, and 61435011).

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References

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  1. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
    [Crossref]
  2. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y.-X. 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]
  3. 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]
  4. 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]
  5. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
    [Crossref]
  6. V. D’Ambrosio, F. Cardano, E. Karimi, E. Nagali, E. Santamato, L. Marrucci, and F. Sciarrino, “Test of mutually unbiased bases for six-dimensional photonic quantum systems,” Sci. Rep. 3, 2726 (2013).
    [PubMed]
  7. 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]
  8. 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]
  9. K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
    [Crossref]
  10. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
    [Crossref]
  11. V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
    [PubMed]
  12. M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
    [Crossref] [PubMed]
  13. Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, and B.-S. Shi, “Optical vortex beam based optical fan for high-precision optical measurements and optical switching,” Opt. Lett. 39(17), 5098–5101 (2014).
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  14. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
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  15. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
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  16. A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
    [Crossref]
  17. D. S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
    [Crossref] [PubMed]
  18. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic-generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56(5), 4193–4196 (1997).
    [Crossref]
  19. Z.-Y. Zhou, D.-S. Ding, Y.-K. Jiang, Y. Li, S. Shi, X.-S. Wang, and B.-S. Shi, “Orbital angular momentum light frequency conversion and interference with quasi-phase matching crystals,” Opt. Express 22(17), 20298–20310 (2014).
    [Crossref] [PubMed]
  20. Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, B.-S. Shi, and G.-C. Guo, “Highly efficient second harmonic generation of a light carrying orbital angular momentum in an external cavity,” Opt. Express 22(19), 23673–23678 (2014).
    [Crossref] [PubMed]
  21. Y. Li, Z.-Y. Zhou, D.-S. Ding, and B.-S. Shi, “Sum frequency generation with two orbital angular momentum carrying laser beams,” J. Opt. Soc. Am. B 32(3), 407–411 (2015).
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  22. Y. Li, Z.-Y. Zhou, D.-S. Ding, W. Zhang, S. Shi, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum photonic quantum interface,” Light: Sci. Appl. (submitted to).
  23. R. Tang, X. Li, W. Wu, H. Pan, H. Zeng, and E. Wu, “High efficiency frequency upconversion of photons carrying orbital angular momentum for a quantum information interface,” Opt. Express 23(8), 9796–9802 (2015).
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  24. G.-H. Shao, Z.-J. Wu, J.-H. Chen, F. Xu, and Y.-Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase-matching,” Phys. Rev. A 88(6), 063827 (2013).
    [Crossref]
  25. W. T. Buono, L. F. C. Moraes, J. A. O. Huguenin, C. E. R. Souza, and A. Z. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16(9), 093041 (2014).
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  26. Q.-F. Chen, B.-S. Shi, Y.-S. Zhang, and G.-C. Guo, “Entanglement of orbital angular momentum states of photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78(5), 053810 (2008).
    [Crossref]
  27. D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Image transfer through two sequential four-wave-mixing processes in hot atomic vapour,” Phys. Rev. A 85(5), 053815 (2012).
    [Crossref]
  28. G. Walker, A. S. Arnold, and S. Franke-Arnold, “Trans-spectral orbital angular momentum transfer via four-wave mixing in Rb vapor,” Phys. Rev. Lett. 108(24), 243601 (2012).
    [Crossref] [PubMed]
  29. G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
    [Crossref] [PubMed]
  30. B. Jack, A. M. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. G. Ireland, S. M. Barnett, and M. J. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A 81(4), 043844 (2010).
    [Crossref]
  31. J. Romero, J. Leach, B. Jack, S. M. Barnett, M. J. Padgett, and S. Franke-Arnold, “Violation of Leggett inequalities in orbital angular momentum subspaces,” New J. Phys. 12(12), 123007 (2010).
    [Crossref]
  32. J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. W. Boyd, A. K. Jha, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express 17(10), 8287–8293 (2009).
    [Crossref] [PubMed]
  33. B. Jack, J. Leach, H. Ritsch, S. M. Barnett, M. J. Padgett, and S. Franke-Arnold, “Precise quantum tomography of photon pairs with entangled orbital angular momentum,” New J. Phys. 11(10), 103024 (2009).
    [Crossref]
  34. R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
    [Crossref] [PubMed]
  35. M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. U.S.A. 111(17), 6243–6247 (2014).
    [Crossref] [PubMed]
  36. S. P. Walborn, A. N. de Oliveira, R. S. Thebaldi, and C. H. Monken, “Entanglement and conservation of orbital angular momentum in spontaneous parametric down-conversion,” Phys. Rev. A 69(2), 023811 (2004).
    [Crossref]
  37. J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A 86(1), 012334 (2012).
    [Crossref]
  38. F. M. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Bounds and optimisation of orbital angular momentum bandwidths within parametric down-conversion systems,” Eur. Phys. J. D 66(7), 178 (2012).
    [Crossref]
  39. H. Di Lorenzo Pires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. 104(2), 020505 (2010).
    [Crossref] [PubMed]
  40. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24(7), 430–432 (1999).
    [Crossref] [PubMed]
  41. G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108(19), 190401 (2012).
    [Crossref] [PubMed]
  42. 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]
  43. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23(15), 880–884 (1969).
    [Crossref]
  44. S. J. Freedman and J. F. Clauser, “Experimental test of local hidden-variable theories,” Phys. Rev. Lett. 28(14), 938–941 (1972).
    [Crossref]
  45. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
    [Crossref]

2015 (2)

2014 (7)

W. T. Buono, L. F. C. Moraes, J. A. O. Huguenin, C. E. R. Souza, and A. Z. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16(9), 093041 (2014).
[Crossref]

Z.-Y. Zhou, D.-S. Ding, Y.-K. Jiang, Y. Li, S. Shi, X.-S. Wang, and B.-S. Shi, “Orbital angular momentum light frequency conversion and interference with quasi-phase matching crystals,” Opt. Express 22(17), 20298–20310 (2014).
[Crossref] [PubMed]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, B.-S. Shi, and G.-C. Guo, “Highly efficient second harmonic generation of a light carrying orbital angular momentum in an external cavity,” Opt. Express 22(19), 23673–23678 (2014).
[Crossref] [PubMed]

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[Crossref] [PubMed]

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. U.S.A. 111(17), 6243–6247 (2014).
[Crossref] [PubMed]

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]

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, and B.-S. Shi, “Optical vortex beam based optical fan for high-precision optical measurements and optical switching,” Opt. Lett. 39(17), 5098–5101 (2014).
[Crossref] [PubMed]

2013 (6)

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[PubMed]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

V. D’Ambrosio, F. Cardano, E. Karimi, E. Nagali, E. Santamato, L. Marrucci, and F. Sciarrino, “Test of mutually unbiased bases for six-dimensional photonic quantum systems,” Sci. Rep. 3, 2726 (2013).
[PubMed]

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]

D. S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
[Crossref] [PubMed]

G.-H. Shao, Z.-J. Wu, J.-H. Chen, F. Xu, and Y.-Q. Lu, “Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase-matching,” Phys. Rev. A 88(6), 063827 (2013).
[Crossref]

2012 (7)

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Image transfer through two sequential four-wave-mixing processes in hot atomic vapour,” Phys. Rev. A 85(5), 053815 (2012).
[Crossref]

G. Walker, A. S. Arnold, and S. Franke-Arnold, “Trans-spectral orbital angular momentum transfer via four-wave mixing in Rb vapor,” Phys. Rev. Lett. 108(24), 243601 (2012).
[Crossref] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y.-X. 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]

J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A 86(1), 012334 (2012).
[Crossref]

F. M. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Bounds and optimisation of orbital angular momentum bandwidths within parametric down-conversion systems,” Eur. Phys. J. D 66(7), 178 (2012).
[Crossref]

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108(19), 190401 (2012).
[Crossref] [PubMed]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
[Crossref] [PubMed]

2011 (4)

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]

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
[Crossref]

2010 (4)

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref] [PubMed]

B. Jack, A. M. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. G. Ireland, S. M. Barnett, and M. J. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A 81(4), 043844 (2010).
[Crossref]

J. Romero, J. Leach, B. Jack, S. M. Barnett, M. J. Padgett, and S. Franke-Arnold, “Violation of Leggett inequalities in orbital angular momentum subspaces,” New J. Phys. 12(12), 123007 (2010).
[Crossref]

H. Di Lorenzo Pires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. 104(2), 020505 (2010).
[Crossref] [PubMed]

2009 (2)

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. W. Boyd, A. K. Jha, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express 17(10), 8287–8293 (2009).
[Crossref] [PubMed]

B. Jack, J. Leach, H. Ritsch, S. M. Barnett, M. J. Padgett, and S. Franke-Arnold, “Precise quantum tomography of photon pairs with entangled orbital angular momentum,” New J. Phys. 11(10), 103024 (2009).
[Crossref]

2008 (1)

Q.-F. Chen, B.-S. Shi, Y.-S. Zhang, and G.-C. Guo, “Entanglement of orbital angular momentum states of photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78(5), 053810 (2008).
[Crossref]

2007 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

2004 (2)

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]

S. P. Walborn, A. N. de Oliveira, R. S. Thebaldi, and C. H. Monken, “Entanglement and conservation of orbital angular momentum in spontaneous parametric down-conversion,” Phys. Rev. A 69(2), 023811 (2004).
[Crossref]

2002 (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[Crossref]

2001 (2)

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]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
[Crossref]

1999 (1)

1997 (1)

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic-generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56(5), 4193–4196 (1997).
[Crossref]

1992 (1)

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]

1972 (1)

S. J. Freedman and J. F. Clauser, “Experimental test of local hidden-variable theories,” Phys. Rev. Lett. 28(14), 938–941 (1972).
[Crossref]

1969 (1)

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23(15), 880–884 (1969).
[Crossref]

Ahmed, N.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y.-X. 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.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108(19), 190401 (2012).
[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]

Allen, L.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic-generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56(5), 4193–4196 (1997).
[Crossref]

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]

Andersson, E.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
[Crossref]

Aolita, L.

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
[PubMed]

Arnold, A. S.

G. Walker, A. S. Arnold, and S. Franke-Arnold, “Trans-spectral orbital angular momentum transfer via four-wave mixing in Rb vapor,” Phys. Rev. Lett. 108(24), 243601 (2012).
[Crossref] [PubMed]

Barnett, S.

Barnett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

J. Romero, D. Giovannini, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A 86(1), 012334 (2012).
[Crossref]

F. M. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Bounds and optimisation of orbital angular momentum bandwidths within parametric down-conversion systems,” Eur. Phys. J. D 66(7), 178 (2012).
[Crossref]

B. Jack, A. M. Yao, J. Leach, J. Romero, S. Franke-Arnold, D. G. Ireland, S. M. Barnett, and M. J. Padgett, “Entanglement of arbitrary superpositions of modes within two-dimensional orbital angular momentum state spaces,” Phys. Rev. A 81(4), 043844 (2010).
[Crossref]

J. Romero, J. Leach, B. Jack, S. M. Barnett, M. J. Padgett, and S. Franke-Arnold, “Violation of Leggett inequalities in orbital angular momentum subspaces,” New J. Phys. 12(12), 123007 (2010).
[Crossref]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref] [PubMed]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. W. Boyd, A. K. Jha, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express 17(10), 8287–8293 (2009).
[Crossref] [PubMed]

B. Jack, J. Leach, H. Ritsch, S. M. Barnett, M. J. Padgett, and S. Franke-Arnold, “Precise quantum tomography of photon pairs with entangled orbital angular momentum,” New J. Phys. 11(10), 103024 (2009).
[Crossref]

Beijersbergen, M. W.

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]

Bowman, R.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

Boyd, R. W.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[Crossref] [PubMed]

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329(5992), 662–665 (2010).
[Crossref] [PubMed]

J. Leach, B. Jack, J. Romero, M. Ritsch-Marte, R. W. Boyd, A. K. Jha, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Violation of a Bell inequality in two-dimensional orbital angular momentum state-spaces,” Opt. Express 17(10), 8287–8293 (2009).
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J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y.-X. 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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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
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M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. U.S.A. 111(17), 6243–6247 (2014).
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R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338(6107), 640–643 (2012).
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A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
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Zeng, H.

Zhang, W.

Zhang, Y.-S.

Q.-F. Chen, B.-S. Shi, Y.-S. Zhang, and G.-C. Guo, “Entanglement of orbital angular momentum states of photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78(5), 053810 (2008).
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Zhou, Z.-Y.

Zou, X.-B.

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Image transfer through two sequential four-wave-mixing processes in hot atomic vapour,” Phys. Rev. A 85(5), 053815 (2012).
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Eur. Phys. J. D (1)

F. M. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Bounds and optimisation of orbital angular momentum bandwidths within parametric down-conversion systems,” Eur. Phys. J. D 66(7), 178 (2012).
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J. Opt. Soc. Am. B (1)

Nat. Commun. (2)

V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
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D. S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
[Crossref] [PubMed]

Nat. Photonics (3)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y.-X. 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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K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
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M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
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Nat. Phys. (2)

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A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
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Nature (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).
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New J. Phys. (3)

J. Romero, J. Leach, B. Jack, S. M. Barnett, M. J. Padgett, and S. Franke-Arnold, “Violation of Leggett inequalities in orbital angular momentum subspaces,” New J. Phys. 12(12), 123007 (2010).
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B. Jack, J. Leach, H. Ritsch, S. M. Barnett, M. J. Padgett, and S. Franke-Arnold, “Precise quantum tomography of photon pairs with entangled orbital angular momentum,” New J. Phys. 11(10), 103024 (2009).
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W. T. Buono, L. F. C. Moraes, J. A. O. Huguenin, C. E. R. Souza, and A. Z. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16(9), 093041 (2014).
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Opt. Express (5)

Opt. Lett. (2)

Phys. Rev. A (9)

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Figures (4)

Fig. 1
Fig. 1 Experimental setup. The simplified upper diagram shows the information transfer direction. M1–M9: mirrors; VPP: vortex phase plate; F1–F3: filters; DM: dichromatic mirror; PPKTP1, 2: periodically poled KTP crystals; SMF1, 2: single mode fibers; L1–L3:150-mm lenses; L4, L7:200-mm lenses; L5, L6:100-mm lenses; PZT: piezoelectric transducer; SLMA, SLMB: spatial light modulators.
Fig. 2
Fig. 2 Experimental results for measurementof the orbital angular momentum(OAM) correlation between the signal and idler photons for different pump OAMs at 1550nm. (a) and (c) show the coincidence counts in 10 s for signal and idler OAMs ranging from −5 to 5 for pump OAMs of 0 and−1, respectively; (b) and (d) show the coincidence counts when the signal and idler OAMs sum to give pump OAMs of 0 and −1, respectively.
Fig. 3
Fig. 3 Experimental results for characterizationof the entanglement in two-dimensional subspaces.(a) and (b) show the coincidence counts of the signal and the idler as a function of phase mask angle θ A when the phase mask angle θ B =0,π/4l (l = 1, 2); (c) and (d) show the real and imaginary parts of the reconstructed density matrix for |Φ 2 .
Fig. 4
Fig. 4 Holograms used in the experimental measurements. (a) and (b) are the holograms for the single OAM states that were used in the measurements of Fig. 1; (c) and (d) show the definitions of θ A and θ B ; (e)–(h) show holograms that were used for quantum tomography measurements.

Equations (6)

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|Φ= m s = m s =+ c m s | m s , λ s | l p m s , λ i .
|θ= 1 2 ( e ilθ |l+ e ilθ | l ).
C( θ A , θ B )= | θ A | θ B ||Φ | 2 cos 2 [l( θ A θ B )].
|Φ l = 1 2 (| l,795 | l,1550 +| l,795 | l,1550 ).
S=E( θ A , θ B )E( θ A , θ B )+E( θ A , θ B )+E( θ A , θ B ).
E( θ A , θ B )= C( θ A , θ B )+C( θ A + π 2l , θ B + π 2l )C( θ A + π 2l , θ B )C( θ A , θ B + π 2l ) C( θ A , θ B )+C( θ A + π 2l , θ B + π 2l )+C( θ A + π 2l , θ B )+C( θ A , θ B + π 2l ) .

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