Abstract

It is shown that orbital angular momentum (OAM) is a promising new resource in future classical and quantum communications. However, the separation of OAM modes is still a big challenge. In this paper, we propose a simple and efficient separation method with a radial varying phase. In the method, specific radial varying phases are designed and modulated for different OAM modes. The resultant beam is focused to the spots with different horizontal and vertical positions after a convex lens, when the coordinate transformation, including two optical elements with coordinate transformation phase and correct phase, operates on the received beam. The horizontal position of the spot is determined by the vortex phases, and the vertical position of the spot is dependent on the radial varying phases. The simulation and experimental results show that the proposed method is feasible both for separation of two OAM modes and separation of three OAM modes. The proposed separation method is available in principle for any neighboring OAM modes because the radial varying phase is controlled. Additionally, no extra instruments are introduced, and there is no diffraction and narrowing process limitation for the separation.

© 2017 Chinese Laser Press

1. INTRODUCTION

Recently, considerable attention has been paid to orbital angular momentum (OAM) modes because of the dramatic increase of information capacity for optical communication [1,2] and the potential of increased bandwidth for quantum cryptography [3]. It has been demonstrated that the information transmission rate can even increase to more than 100 Tb/s by multiplexing with 12 OAM modes, two polarizations, and 42 wavelengths [4]. The efficient separation of OAM modes plays an important role in OAM applications.

The simplest method for measuring the OAM content of a beam is to perform a series of projection measurements [1,5], where an OAM mode with topological charge is first transferred to a flat phase beam by being illuminated on a forked hologram with and then detected by a power detector [6,7]. Later, Leach et al. presented a technique for separating OAM modes by using a Mach–Zehnder interferometer at the single-photon level [8]. To efficiently sort N OAM modes simultaneously, Berkhout et al. proposed a separating method based on the transformation from Cartesian to log-polar [9]. In addition, simulations and experiments have demonstrated that the separating method has distinguished different OAM modes simultaneously with a detector array [10,11]. Mirhosseini et al. presented a better OAM separation method with a fan-out technique to overcome the diffraction limitation [12]. Lavery et al. measure an OAM spectrum that distributes in both the horizontal and vertical dimensions by changing the radial of a beam [13]. However, the separation of OAM mode is still challenging; there is yet no method as easy as the separation of two polarization modes.

In this paper, we propose a highly efficient separation of OAM modes method with radial varying phase. A special radial varying phase is designed and added for each separated OAM mode. The coordinate transformation is afterward operated on the received beam. The received beam is unfolded, and different spots with different horizontal and vertical positions are obtained with a focused lens, where the horizontal position is determined by the vortex phases, and the vertical position is dependent on the radial varying phase. The focal spots are both separated in the horizontal and vertical dimensions; importantly, the vertical position can be controlled.

This paper is organized as follows. In Section 2, we describe the proposed separation method with radial varying phase. In Section 3, we present the numerical and experimental results of the proposed method. Finally, we draw a conclusion in Section 4.

2. OAM SEPARATION WITH RADIAL VARYING PHASE

The concept of the proposed method is shown in Fig. 1. The electric field of an OAM mode with topological charge is represented by

U0(r,θ)=A(r)exp(iθ),
where A(r)exp(r2/w2) is the amplitude function of the beam, r is the radial index, w is waist size, and θ is the azimuthal index. The frame of a beam of OAM generates an OAM mode beam with a spatial light modulator (SLM). A special radial varying phase for each OAM mode is successively designed by radial varying phase and added to the OAM mode beam. Here, the radial varying phase with parameter m is designed as
ϕ1(r)={m2πlnrlnR(Rmin<r<Rmax)0(others),
where R is a constant related to the maximum of the beam size, Rmax is the radius of the SLM size, and Rmin is the inner radius of the OAM doughnut. For simplification, Rmin is always set up to one pixel size. m denotes the spatial frequency of the phase. lnr is used for representing nonlinear property of the phase.

 figure: Fig. 1.

Fig. 1. OAM separation method with radial varying phase. U0(r,θ),,U5(x,y) are the electric fields, l1,l2 denote different OAM modes, and m1,m2 represent different radial varying phases. Image e is the superposition beam. Image f is the unfolded beam from image e by the coordinate transformation. FSO: free space optical channel. Images a and c are the horizontal distribution phases that unfolded from the vortex phases, and images b and d are the vertical distribution phases unfolded from the radial varying phases.

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Now, the electric field of the OAM beam becomes

U1(r,θ)=U0(r,θ)exp[iϕ1(r)].
Furthermore, the superposition of multiple OAM modes with different radial varying phases is expressed as
U2(r,θ)=,mA(r)exp(iθ)exp(im2πlnrlnR).

The superposition beam arrives at the receiver after propagating some distance in the optical free-space channel. The frame of coordinate transformation operates coordinate transformation on the receiving superposition beam, transforming the beam from Cartesian to log-polar coordinates. There are two optical phase elements, together with a Fourier lens (FT), in coordinate transformation. The first optical element has transformation phase ϕ2(x,y), where ϕ2(x,y)=2aπfλ[yarctan(yx)xln(x2+y2b)+x]; a and b are scaling constants [9]. When the superposition beam passes through the first optical element followed by the FT lens, the input plane (x,y) is mapped to log-polar plane (u,v), that is,

U3(u,v)=U2(r,θ)exp[iϕ2(x,y)]exp(ikxv+yuf)dxdy,
where x=rcosθ, y=rsinθ, k=2π/λ, and f is the focal length of the FT. The position and shape of beam in the output plane (u,v) is dependent on ϕ2(x,y) [14], where v=aarctan(y/x), and u=aln(x2+y2/b).

This transformation introduces some distortions, which can be corrected by the second optical element [9]. The corrected phase ϕ3(u,v) is expressed as 2abπfλexp(ua)cos(va). After the correction, the output electric field U4(u,v) can be given as

U4(u,v)=U3(u,v)exp[iϕ3(u,v)].

In fact, Eq. (6) can be rewritten as the sum of some truncated plane waves according to the relationship between (x,y) and (u,v) [15]:

U4(u,v)=,mrect(v2πa)rect(ualnR)exp(iva)exp(im2πualnR),
where exp(iva) denotes the topological charge influencing the horizontal phase gradient in horizontal axis v, and exp(im2πualnR) denotes the parameter m influencing the vertical phase gradient in vertical axis u.

After a convex lens, the unfolded beam with linear phase is focused into spots. The electric field is further described as

U5(x,y)=U4(u,v)·exp(ikxv+yuf)dudv,
where x and y denote the coordinate position in the focal plane. Since rect(x)exp(i2πxw)d(x) is equal to sincw, Eq. (8) can be rewritten as
U5(x,y)=,m(2πa)sinc(xΔΔ)(alnR)sinc(yΛmΛ),
where Δ is fλ/(2πa), and Λ is fλ/(alnR). With the property of the function sincx, the horizontal and vertical positions of the spots can be obtained as
x=Δ,y=Λm,
which indicates that the spot changes as a function of and m, and the position of the spot in the x direction is influenced by l, while the position of the spot in the y direction is influenced by m.

In Fig. 1, image e shows the Cartesian coordinate before the transformation; image f is the unfolded form of image e after coordinate transformation, which can be decomposed from images a to d. Because the focal positions of each spot can be controlled in the vertical direction by adding a specific radial varying phase, in principle, any two spots corresponding to neighboring OAM modes can be separated.

3. SIMULATION AND EXPERIMENTAL RESULTS

In this section, we test the proposed separation method by simulation and experiment.

The experimental setup is shown in Fig. 2. The beam from an He–Ne laser source (632.8nm) is expanded to a collimated light by passing through lenses L1 (50 mm) and L2 (100 mm), and then is split into two beams by a beam splitter. The two beams individually illuminate one part of SLM1 (620 to 1100 nm), A and B. Each part is loaded with phases, including a vortex phase and a radial varying phase. Then two apertures, A1 and A2, are used to select the first order of the beam to propagate. The multiplexed beam propagates about 2 m in the optical free-space channel. The received beam is unfolded to a rectangle profile plane beam after illuminating SLM2, an FT and SLM3, where SLM2 is loaded with the transformation phase ϕ2(x,y), and SLM3 is loaded with the correction phase ϕ3(u,v). The resultant beam is focused into spots by a convex lens (L5), and a CCD is located in the focal plane of the convex lens.

 figure: Fig. 2.

Fig. 2. Schematic setup for the experiment. L1–L5, lenses; BS, beam splitter; M1 and M2, mirrors; A1 and A2, apertures.

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First, the property of the OAM mode with radial varying phase is demonstrated in Fig. 3. The multiplexed modes are OAM3 and OAM+5, where OAM3 is loaded with m=0,2,and+2 radial varying phases, while OAM+5 does not have radial varying phase. When OAM3 is loaded with radial varying phase, the interferogram displays the clockwise rotation for m=2, and counterclockwise rotation for m=2. It is shown that the multiplexed mode is controlled by the radial varying phase m.

 figure: Fig. 3.

Fig. 3. Demonstration of the propagation properties of the radial varying phase.

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Figure 4 shows the simulation and experimental results of the proposed method with OAM3 and OAM+5 and the average separation efficiency. Two parameters m1,m2 represent the radial varying phases for OAM3 and OAM+5, respectively. The first row denotes the results without the radial varying phases (m1=m2=0), and the second row represents OAM3 being loaded with m1=2 radial varying phase and OAM+5 without radial varying phase. The results show focal spots are distributed in only the horizontal direction without radial varying phases, and the focal spot of OAM3 is moved down in vertical direction with m1=2. The position of the focal spot is determined both by OAM topological charges l and radial varying phases m.

 figure: Fig. 4.

Fig. 4. Simulation and experimental results for the proposed separation method with two OAM modes.

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Figure 5 further demonstrates the simulation and experiment results of the proposed method for three OAM modes: OAM5, OAM1, and OAM+5 and their separation efficiencies. The parameters mi(i=1,2,and3) represent the radial varying phases for OAMi(i=5,1,and5), respectively. In the first column, parameters of mi are 0. This means that there are no radial varying phases loaded on the corresponding OAM modes. Three spots are at the same horizontal positions. In the second column, only OAM5 is loaded with m1=2 and the spot of OAM5 mode is moved down. In the third column, the spots of OAM5 and OAM+5 are moved down when loaded with the same radial varying phases m1=m3=2. In the fourth column, the spot of OAM+5 is moved up for m3=2. OAM1 is kept on the original because of no radial varying phase. The results further show that the vertical position of the spot is determined by the parameter m, where m can be designed and controlled.

 figure: Fig. 5.

Fig. 5. Simulation and experimental results for the proposed separation method with three OAM modes.

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To compare with the existing separation method using coordinate transformation, Fig. 6 shows the simulation results using the methods in Refs. [9,12] and the proposed method for the neighboring modes OAM3 and OAM2. The results show that the neighboring OAM modes overlap with the method in Ref. [9], the overlapping is decreased with the method in Ref. [12], and no overlapping exists with the proposed separation method. For Fig. 6(c), the radial varying phase is m=+3. If there is still some overlapping between the neighboring OAM modes, the radial varying phase m for OAM3 could be set to a large number to separate the two neighboring OAM modes completely. The limitation of the proposed method is that the designed varying phase should be modulated at each OAM modes at first.

 figure: Fig. 6.

Fig. 6. Comparisons of the separation methods based on coordinate transformation and their corresponding separation efficiency. (a) illustrates the method in Ref. [9]; (b) illustrates the method in Ref. [12]; (c) illustrates the proposed method.

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4. CONCLUSION

In this paper, we have proposed a simple and efficient OAM separation method with radial varying phase, where the radial varying phase is labeled by the parameter m. OAM modes have been modulated by different radial varying phases. For the separation, the coordinate transformation has transferred the superposition OAM mode beam to spots with different horizontal and vertical positions, where the horizontal and vertical positions of the spot are determined by the OAM topological charge and the radial varying phase, respectively. The simulation and experimental results have satisfied the theoretical ones. Since the radial varying phase could be added with the same optical device as the vortex phase, the proposed method has not introduced any extra instruments. Importantly, the separation for any neighboring OAM modes is available because the radial varying phase is controlled. Furthermore, the proposed separation method provides a novel idea to avoid the diffraction limitations on OAM separation.

Funding

National Natural Science Foundation of China (NSFC) (61475075, 61271238); Open Research Fund of Key Lab of Broadband Wireless Communication and Sensor Network Technology, Ministry of Education of the People's Republic of China (MOE) (NYKL2015011).

REFERENCES

1. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, and Y. Yan, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012). [CrossRef]  

2. L. Li, G. Xie, Y. Ren, N. Ahmed, and H. Huang, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55, 2098–2103 (2016). [CrossRef]  

3. A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002). [CrossRef]  

4. H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39, 197–200 (2014). [CrossRef]  

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

6. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, and V. Pas’ko, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004). [CrossRef]  

7. R. Tudora, M. Mihailescuc, C. Kuskoa, I. Paunc, and A. Nanc, “Simultaneous and spatially separated detection of multiple orbital angular momentum states,” Opt. Commun. 368, 141–149 (2016). [CrossRef]  

8. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004). [CrossRef]  

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

10. C. Li, R. Jiang, L. Wang, and S. Zhao, “Simulations of high efficient separation of orbital-angular-momentum of light,” J. Nanjing Univ. Post Telecommun. 36, 4752 (2016).

11. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, and M. J. Padgett, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012). [CrossRef]  

12. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013). [CrossRef]  

13. M. P. J. Lavery, D. J. Robertson, A. Sponselli, and J. Courtial, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013). [CrossRef]  

14. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974). [CrossRef]  

15. M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20, 24444–24449 (2012). [CrossRef]  

References

  • View by:

  1. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, and Y. Yan, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
    [Crossref]
  2. L. Li, G. Xie, Y. Ren, N. Ahmed, and H. Huang, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55, 2098–2103 (2016).
    [Crossref]
  3. A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
    [Crossref]
  4. H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100  Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39, 197–200 (2014).
    [Crossref]
  5. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [Crossref]
  6. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, and V. Pas’ko, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
    [Crossref]
  7. R. Tudora, M. Mihailescuc, C. Kuskoa, I. Paunc, and A. Nanc, “Simultaneous and spatially separated detection of multiple orbital angular momentum states,” Opt. Commun. 368, 141–149 (2016).
    [Crossref]
  8. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
    [Crossref]
  9. G. C. Berkhout, M. P. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
    [Crossref]
  10. C. Li, R. Jiang, L. Wang, and S. Zhao, “Simulations of high efficient separation of orbital-angular-momentum of light,” J. Nanjing Univ. Post Telecommun. 36, 4752 (2016).
  11. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, and M. J. Padgett, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012).
    [Crossref]
  12. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
    [Crossref]
  13. M. P. J. Lavery, D. J. Robertson, A. Sponselli, and J. Courtial, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
    [Crossref]
  14. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
    [Crossref]
  15. M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20, 24444–24449 (2012).
    [Crossref]

2016 (3)

L. Li, G. Xie, Y. Ren, N. Ahmed, and H. Huang, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55, 2098–2103 (2016).
[Crossref]

R. Tudora, M. Mihailescuc, C. Kuskoa, I. Paunc, and A. Nanc, “Simultaneous and spatially separated detection of multiple orbital angular momentum states,” Opt. Commun. 368, 141–149 (2016).
[Crossref]

C. Li, R. Jiang, L. Wang, and S. Zhao, “Simulations of high efficient separation of orbital-angular-momentum of light,” J. Nanjing Univ. Post Telecommun. 36, 4752 (2016).

2014 (1)

2013 (2)

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref]

M. P. J. Lavery, D. J. Robertson, A. Sponselli, and J. Courtial, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[Crossref]

2012 (3)

2010 (1)

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

2004 (2)

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, and V. Pas’ko, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref]

2002 (1)

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

2001 (1)

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

1974 (1)

Ahmed, N.

Barnett, S. M.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref]

Beijersbergen, M. W.

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

Berkhout, G. C.

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

Berkhout, G. C. G.

Birnbaum, K. M.

Boyd, R. W.

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref]

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20, 24444–24449 (2012).
[Crossref]

Bryngdahl, O.

Courtial, J.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, and J. Courtial, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[Crossref]

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

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, and V. Pas’ko, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref]

Dolinar, S. J.

Erkmen, B. I.

Fazal, I. M.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, and Y. Yan, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Gibson, G.

Huang, H.

Jiang, R.

C. Li, R. Jiang, L. Wang, and S. Zhao, “Simulations of high efficient separation of orbital-angular-momentum of light,” J. Nanjing Univ. Post Telecommun. 36, 4752 (2016).

Kuskoa, C.

R. Tudora, M. Mihailescuc, C. Kuskoa, I. Paunc, and A. Nanc, “Simultaneous and spatially separated detection of multiple orbital angular momentum states,” Opt. Commun. 368, 141–149 (2016).
[Crossref]

Lavery, M. P.

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

Lavery, M. P. J.

Leach, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref]

Li, C.

C. Li, R. Jiang, L. Wang, and S. Zhao, “Simulations of high efficient separation of orbital-angular-momentum of light,” J. Nanjing Univ. Post Telecommun. 36, 4752 (2016).

Li, L.

Love, G. D.

Mair, A.

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

Malik, M.

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref]

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20, 24444–24449 (2012).
[Crossref]

Mihailescuc, M.

R. Tudora, M. Mihailescuc, C. Kuskoa, I. Paunc, and A. Nanc, “Simultaneous and spatially separated detection of multiple orbital angular momentum states,” Opt. Commun. 368, 141–149 (2016).
[Crossref]

Mirhosseini, M.

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref]

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express 20, 24444–24449 (2012).
[Crossref]

Nanc, A.

R. Tudora, M. Mihailescuc, C. Kuskoa, I. Paunc, and A. Nanc, “Simultaneous and spatially separated detection of multiple orbital angular momentum states,” Opt. Commun. 368, 141–149 (2016).
[Crossref]

O’Sullivan, M. N.

Padgett, M. J.

Pas’ko, V.

Paunc, I.

R. Tudora, M. Mihailescuc, C. Kuskoa, I. Paunc, and A. Nanc, “Simultaneous and spatially separated detection of multiple orbital angular momentum states,” Opt. Commun. 368, 141–149 (2016).
[Crossref]

Ren, Y.

Robertson, D. J.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, and J. Courtial, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[Crossref]

M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, and M. J. Padgett, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012).
[Crossref]

Rogawski, D.

Shi, Z.

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref]

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref]

Sponselli, A.

M. P. J. Lavery, D. J. Robertson, A. Sponselli, and J. Courtial, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[Crossref]

Tudora, R.

R. Tudora, M. Mihailescuc, C. Kuskoa, I. Paunc, and A. Nanc, “Simultaneous and spatially separated detection of multiple orbital angular momentum states,” Opt. Commun. 368, 141–149 (2016).
[Crossref]

Tur, M.

Vasnetsov, M.

Vaziri, A.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

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

Wang, J.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, and Y. Yan, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Wang, L.

C. Li, R. Jiang, L. Wang, and S. Zhao, “Simulations of high efficient separation of orbital-angular-momentum of light,” J. Nanjing Univ. Post Telecommun. 36, 4752 (2016).

Weihs, G.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

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

Willner, A. E.

Willner, M. J.

Xie, G.

Yan, Y.

Yang, J. Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, and Y. Yan, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Yue, Y.

Zeilinger, A.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

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

Zhao, S.

C. Li, R. Jiang, L. Wang, and S. Zhao, “Simulations of high efficient separation of orbital-angular-momentum of light,” J. Nanjing Univ. Post Telecommun. 36, 4752 (2016).

Appl. Opt. (1)

J. Nanjing Univ. Post Telecommun. (1)

C. Li, R. Jiang, L. Wang, and S. Zhao, “Simulations of high efficient separation of orbital-angular-momentum of light,” J. Nanjing Univ. Post Telecommun. 36, 4752 (2016).

J. Opt. Soc. Am. (1)

Nat. Commun. (1)

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref]

Nat. Photonics (1)

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, and Y. Yan, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Nature (1)

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

New J. Phys. (1)

M. P. J. Lavery, D. J. Robertson, A. Sponselli, and J. Courtial, “Efficient measurement of an optical orbital-angular-momentum spectrum comprising more than 50 states,” New J. Phys. 15, 013024 (2013).
[Crossref]

Opt. Commun. (1)

R. Tudora, M. Mihailescuc, C. Kuskoa, I. Paunc, and A. Nanc, “Simultaneous and spatially separated detection of multiple orbital angular momentum states,” Opt. Commun. 368, 141–149 (2016).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. Lett. (3)

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[Crossref]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92, 013601 (2004).
[Crossref]

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

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Figures (6)

Fig. 1.
Fig. 1. OAM separation method with radial varying phase. U0(r,θ),,U5(x,y) are the electric fields, l1,l2 denote different OAM modes, and m1,m2 represent different radial varying phases. Image e is the superposition beam. Image f is the unfolded beam from image e by the coordinate transformation. FSO: free space optical channel. Images a and c are the horizontal distribution phases that unfolded from the vortex phases, and images b and d are the vertical distribution phases unfolded from the radial varying phases.
Fig. 2.
Fig. 2. Schematic setup for the experiment. L1–L5, lenses; BS, beam splitter; M1 and M2, mirrors; A1 and A2, apertures.
Fig. 3.
Fig. 3. Demonstration of the propagation properties of the radial varying phase.
Fig. 4.
Fig. 4. Simulation and experimental results for the proposed separation method with two OAM modes.
Fig. 5.
Fig. 5. Simulation and experimental results for the proposed separation method with three OAM modes.
Fig. 6.
Fig. 6. Comparisons of the separation methods based on coordinate transformation and their corresponding separation efficiency. (a) illustrates the method in Ref. [9]; (b) illustrates the method in Ref. [12]; (c) illustrates the proposed method.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

U0(r,θ)=A(r)exp(iθ),
ϕ1(r)={m2πlnrlnR(Rmin<r<Rmax)0(others),
U1(r,θ)=U0(r,θ)exp[iϕ1(r)].
U2(r,θ)=,mA(r)exp(iθ)exp(im2πlnrlnR).
U3(u,v)=U2(r,θ)exp[iϕ2(x,y)]exp(ikxv+yuf)dxdy,
U4(u,v)=U3(u,v)exp[iϕ3(u,v)].
U4(u,v)=,mrect(v2πa)rect(ualnR)exp(iva)exp(im2πualnR),
U5(x,y)=U4(u,v)·exp(ikxv+yuf)dudv,
U5(x,y)=,m(2πa)sinc(xΔΔ)(alnR)sinc(yΛmΛ),
x=Δ,y=Λm,

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