Abstract

Orbital angular momentum (OAM) entangled photons propagating through non-Kolmogorov turbulence are studied by numerical simulations. Here, the paper uses the multiphase screen model, especially focusing on the influences of the azimuthal mode and the turbulence parameters (i.e., the generalized exponent, the outer scale of turbulence, and the inner scale of turbulence) on entanglement evolution in the weak scintillation regime. The results indicate that the azimuthal mode, the generalized exponent, and the outer scale of turbulence have obvious influences on OAM entanglement. However, the influence of the turbulence inner scale on OAM entanglement can be ignored.

© 2016 Optical Society of America

1. INTRODUCTION

The orbital angular momentum (OAM) photon has attracted much attention, because in principle it has an infinite number of eigenstates, which not only can store and process quantum states in a high-dimensional Hilbert space, but also hold promise for many potential applications in quantum information processing [15]. However, when entangled OAM photons propagate across turbulence, the beam wandering, wavefront distortions, and scintillation destroy the coherence of OAM entanglement [610], which is the biggest challenge for realizing OAM-based quantum communication. Therefore, it is more imperative to describe the OAM entanglement evolution in turbulence so the OAM modes can be successfully applied to the free-space quantum communication.

Recently, the effect of atmospheric turbulence on entangled OAM photons has been addressed in many theoretical and experimental studies [7,1120]. Most of them are based on the Kolmogorov power spectrum, which is widely accepted and applied to researching optical wave propagation in the atmosphere. Until now, it has been experimentally shown that, generally, atmospheric turbulence might possess a different structure from the conventional Kolmogorov one [21]. According to several observations, there have been significant deviations from the Kolmogorov turbulence model in the upper troposphere and stratosphere, as well as along the non-homogeneous path [2224]. Besides, to our knowledge, in the Kolmogorov power spectrum, the influence of the inner and outer scales, which may be very important to entanglement evolution, was neglected in most previous studies.

In this paper, we have numerically studied the propagation of two OAM entangled photons through non-Kolmogorov turbulence. Through the use of the multiphase screen model, the influences of the azimuthal mode, the generalized exponent, the turbulent outer scale, and the turbulent inner scale on entanglement evolution in the weak scintillation regime are investigated in detail. Meanwhile, the results drawn are compared with the Smith and Raymer theories (S&R) [16] and the result in Ref. [11]. The paper is organized as follows: the numerical procedure is introduced in Section 2. In Section 3, the numerical results are presented. The conclusion is presented in Section 4.

2. NUMERICAL PROCEDURE

The numerical setup is shown in Fig. 1. Without loss of generality, it is assumed that a source field produces a pair of photons, whose input state is a Bell state encoded by LG modes with the opposite azimuthal quantum number

|ψin=12(|A|B+|A|B),
where the subscripts A and B are labeled as the two different paths and the parameter is the azimuthal mode order. In normalized cylindrical coordinates, the LG modes are expressed as
MpLG=Nr||exp(iφ)(1+it)p(1it)p+||+1LP||(2r21+t2)exp(r21it),
where Lp|| is the generalized Laguerre polynomial, with p being the radial index, r=(x2+y2)1/2/ω0, φ is the azimuthal angle, ω0 is the beam waist radius, t=z/zR with zR is the Rayleigh range (zR=πω02/λ), and λ is the wavelength. The normalization constant is given by
N=(p!2||+1π(p+||)!)1/2.

We first consider the random refractive index fluctuation δn(x), which in an inhomogeneous turbulent medium is presented as

δn(x)=n1,
where n is the refractive index, and x=xx^+yy^+zz^ is a three-dimensional position vector. From Eq. (4) it can be seen that the random refractive index fluctuation δn(x) (δn1) is far less than the refractive index n. Besides, it is assumed that the field propagating through the turbulent atmosphere tends to be paraxial and uniformly polarized. It is found that the propagation of two OAM entangled photons through a turbulent atmosphere can be simulated by a standard split-step method, in which the path L is broken up into N discrete steps. Each phase screen of turbulent atmosphere is replaced by non-turbulent propagation followed by an effective thin phase screen, as shown in Fig. 1. The phase function for each phase screen can be expressed as
φ(x)=k00Δzδn(x)dz,
where k0 is the wave number in a vacuum, and Δz represents the thickness of each phase screen. As seen from Eq. (5), each phase screen introduces random phase errors, which eventually destroy the entanglement.

 

Fig. 1. Source produces a pairs of OAM-entangled qubits, whose wave fronts get deteriorated as they propagate through a series of phase screens along the (horizontal) z axis toward a detector.

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In the following discussions, the paper employs the non-Kolmogorov power spectral for the refractive index fluctuations, given by [22,23]

Φn(κ,z)=A(α)C˜n2(κ2+κ02)α/2exp(κ2/κm2),3<α<4,
where α is the generalized exponent, C˜n2 (in units of mα2) is the generalized refractive-index structure parameter that describes the strength of the turbulence along the path, κ is the magnitude of the three-dimensional wave number vector, κ0=2π/L0, and κm=2π/l0, while L0 and l0 are the outer scale and inner scale of turbulence, respectively. A(α) is defined by [25,26]
A(α)=Γ(α1)cos(απ/2)/(4π2),
where the symbol Γ() is the gamma function. When L0=, l0=0, and C˜n2=Cn2 can be satisfied, Eq. (6) reduces to the conventional Kolmogorov spectrum
Φn(κ)=0.033Cn2κ11/3.

Generally, the turbulent phase screens can be realized by the spectral method [27,28], in which the phase screens are randomly generated by the fast Fourier transform (FFT) associated with the power density spectrum. It is given by

φFFT(m,n)=m=N/2N/21n=N/2N/21h(m,n)Δκ2πk2Φn(κx,κy)Δzei(mm/N+nn/N)ΔκΔκ,
where h is a zero-mean normally distributed random complex-valued function, k is the wave number, and Δκ is the spacing between samples in the frequency domain. Although the FFT-based phase screen is simple and fast, it is deficient in the low-spatial frequency components (0,2π/N), which have an important influence on low-order turbulent effects. Fortunately, the shortcoming can be compensated for by subharmonic methods [29], where the low spatial frequency part of the turbulent phase screen can be expressed as
φLow(m,n)=m=Nx/2Nx/21n=Ny/2Ny/21h(m,n)Δκ2πk2Φn(κx,κy)Δzei(mm/NxN+nn/NyN)ΔκΔκ.

Because the turbulence phase screen is the sum of Eqs. (9) and (10), when Nx and Ny are large, the straightforward evaluation of Eq. (10) is time consuming in the case of no FFT. If Nx=Ny=N is satisfied, then Eq. (10) becomes a fractional Fourier transform, which can be evaluated using the fast algorithm that has a complexity proportional to that of the FFT algorithm [30].

For the Kolmogorov turbulence statistics with a finite outer scale, the phase structure function can be written as [31]

Dφ(r)=6.16r05/3[35(L02π)5/3(rL0/4π)5/6Γ(11/6)K5/6(2πrL0)].
Here, r0 is the Fried parameter, and r is the separation distance. In Fig. 2, for each outer scale, the mean structure function of a sample of 1000 phase screens has been computed, and the results have been compared with the theoretical values given by Eq. (11). As shown in Fig. 2, the values obtained from the simulated phase screens accord well with the theoretical ones, which further implies that the numerical simulations are reliable.

 

Fig. 2. Comparisons between Eq. (11) and statistical mean of the structure function obtained from samples of 1000 phase screens generated with the spectral method. The black color is the results of Eq. (11), while the red represents the sample average results. The value of the Fried parameter is r0=3.18  cm.

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Now let us consider the matrix elements of the output OAM state. Due to the random refractive index fluctuations in a turbulent atmosphere, the matrix elements of the output state are scattered into infinite-dimensional OAM space. However, during the measurement process, only the information that is contained in a finite OAM subspace can be extracted. As a result, the paper has post-selected the transmitted state within the desired OAM subspace (i.e., |A|B, |A|B, |A|B|A|B), and calculated the post-selected density matrices. In addition, to describe the evolution of bi-photon OAM states accurately, the ensemble averages of the post-selected density matrices corresponding to M different instances of the turbulent medium are computed [12,13].

Finally, we normalize the post-selected density matrix and examine the decay of entanglement by calculating the concurrence C(ρ) [32] from the normalized density matrix. The concurrence is plotted as a function, i.e., ω0/ρ0, and ρ0 is the coherence parameter, which can be expressed by [23,33]

ρ0=(A(α)B(α)C˜n2k2L)1α2,
where B(α)=(2)4απ2Γ(2α2)/(2(8α2Γ(2α2))α22Γ(α2)), L is the propagation distance, and k is the optical wave number.

It should be mentioned that the four optical fields are required to simulate each run of the input state. Each optical field is represented by a 256×256 array of samples of the complex-valued function for the mode given in Eq. (2). Furthermore, in the paper, a number of M=500 such runs have been performed to obtain the post-selected density matrices.

3. NUMERICAL RESULTS

First, the numerical result is compared with the S&R theory [16] and the result in Ref. [11]. Here, the paper only considers the case in which both photons pass through atmospheric turbulence with wavelength λ=750  nm, radial index p=0, waist width ω=0.1  m, generalized exponent α=11/3, inner scale l0=5  mm, outer scale L0=500  m, and refraction index structure constant Cn2=2×1015  m2/3. This, of course, means that the evolution of the OAM entanglement is in the weak scintillation regime, where one dimensionless parameter (ω0/ρ0) is required to describe the evolution of the concurrence [13].

As shown in Fig. 3, the numerical result agrees well with the result in Ref. [11] but disagree with the S&R theory curve. This difference comes from the phase structure function employing a quadratic approximation in the S&R theory.

 

Fig. 3. Concurrence as a function of ratio ω0/ρ0 when both the photons propagate through turbulence. In (a) l=1, in (b) l=3, in (c) l=5, and in (d) l=7.

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Now the entanglement evolution of OAM photons propagating in non-Kolmogorov turbulence is investigated by the numerical simulations presented in the previous section. Additionally, the case where both photons pass through atmospheric turbulence is also considered. The results are shown in Figs. 47.

 

Fig. 4. Concurrence as a function of ratio ω0/ρ0 for different values of the azimuthal modes.

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Fig. 5. Concurrence as a function of ratio ω0/ρ0 for different values of the generalized exponent.

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Fig. 6. Concurrence as a function of ratio ω0/ρ0 for different values of the turbulence outer scale with α=3.6.

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Fig. 7. Concurrence as a function of ratio ω0/ρ0 for different values of the turbulence inner scale with α=3.6.

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Figure 4 shows that the concurrence is plotted against the ratio ω0/ρ0 for different values of the azimuthal mode with waist width ω=0.1  m, radial index p=0, and wavelength λ=750  nm. The non-Kolmogorov spectrum parameters are set as α=3.6, l0=5  mm, L0=10  m, and Cn2=2×1015  m3/5. As indicated by Fig. 4, the concurrence decays through the non-Kolmogorov turbulence, but the decay of concurrence becomes slower with the increase of the azimuthal mode, which means the entangled OAM photons with the larger azimuthal mode are less affected by turbulence. This can be understood as follows. With the azimuthal mode increasing, the OAM beam widens, and its phase front oscillates more rapidly, which implies that its spatial phase structure gets finer [11].

Figure 5 shows that the concurrence is plotted against the ratio ω0/ρ0 for different values of the generalized exponent α with ω=0.1  m, λ=750  nm, p=0, =1, l0=5  mm, L0=10  m, and Cn2=2×1015  m3α. As seen from Fig. 5, the concurrence decays slower in turbulence with the increase of the generalized exponent α, thus indicating less influence on the OAM entanglement by the turbulence with the higher generalized exponent α.

In Fig. 6, the concurrence is plotted against the ratio ω0/ρ0 for different values of the turbulence outer scale with ω=0.1  m, λ=750  nm, p=0, =1, l0=5  mm, and Cn2=2×1015  m3α. It is deduced from Fig. 6 that the concurrence can survive longer as the turbulence outer scale decreases. This reveals that the OAM entanglement will be less affected by turbulence with a smaller outer scale. For L0=1  m, shown in Fig. 6, the curve for the concurrence decays as a function of the ratio ω0/ρ0 increases, but at the tail there is a fluctuation behavior, which might be due to fewer statistical samples.

Figure 7 shows the concurrence plotted against the ratio ω0/ρ0 for different values of the turbulence inner scale with ω=0.1  m, λ=750  nm, p=0, =1, L0=10  m, and Cn2=2×1015  m3α. One finds from Fig. 7 that all the decay curves are nearly identical for different values of the turbulence inner scale, which means the influence of the turbulence inner scale on OAM entanglement can be ignored.

4. CONCLUSION

In conclusion, using the multiphase screen model, the paper numerically investigated the entanglement evolution of OAM photons propagating in non-Kolmogorov turbulence, especially focusing on the influences of the azimuthal mode and turbulence parameters on entanglement evolution in the weak scintillation regime. The case in which both photons passed through turbulence was considered. Our numerical results show that the entanglement decays through non-Kolmogorov turbulence, but for larger azimuthal modes, it decays more slowly under the same turbulence condition. Meanwhile, when the other parameters are fixed, the OAM entanglement is less affected by turbulence with a higher generalized exponent or a smaller turbulence outer scale. Moreover, the influence of the turbulence inner scale on OAM entanglement can be ignored. It is believed that these findings may be useful in applications in free-space optical communications.

Funding

Chinese Academy of Sciences (CAS) National Defense Innovation Foundation of China (CXJJ-16S080).

REFERENCES

1. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002). [CrossRef]  

2. M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64, 012306 (2001). [CrossRef]  

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

4. D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre-Gauss modes using the Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008). [CrossRef]  

5. J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008). [CrossRef]  

6. 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, 5448–5456 (2004). [CrossRef]  

7. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005). [CrossRef]  

8. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–230 (2008). [CrossRef]  

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

10. A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E 80, 046609 (2009). [CrossRef]  

11. N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91, 012345 (2015). [CrossRef]  

12. A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013). [CrossRef]  

13. A. H. Ibrahim, F. S. Roux, and T. Konrad, “Parameter dependence in the atmospheric decoherence of modally entangled photon pairs,” Phys. Rev. A 90, 052115 (2014). [CrossRef]  

14. F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011). [CrossRef]  

15. J. R. G. Alonso and T. A. Brun, “Protecting orbital-angular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88, 022326 (2013). [CrossRef]  

16. B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006). [CrossRef]  

17. F. S. Roux, “The Lindblad equation for the decay of entanglement due to atmospheric scintillation,” J. Phys. A 47, 195302 (2014). [CrossRef]  

18. T. Brűnner and F. S. Roux, “Robust entangled qutrit states in atmospheric turbulence,” New J. Phys. 15, 063005 (2013). [CrossRef]  

19. M. V. Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled two-photon beams,” Phys. Rev. A 88, 053836 (2013). [CrossRef]  

20. B.-J. Pros, C. H. Monken, E. R. Elie, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express 19, 6671–6683 (2011). [CrossRef]  

21. D. Dayton, B. Pierson, and B. Spielbusch, “Atmospheric structure function measurements with a Shack-Hartmann wave front sensor,” Opt. Lett. 17, 1737–1739 (1992). [CrossRef]  

22. M. S. Belenkii, S. J. Karis, J. M. Brown II, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997). [CrossRef]  

23. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995). [CrossRef]  

24. D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994). [CrossRef]  

25. H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial Gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28, 1016–1021 (2011). [CrossRef]  

26. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35, 1043–1045 (2010). [CrossRef]  

27. J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988). [CrossRef]  

28. J. S. Xiang, “Fast and accurate simulation of the turbulent phase screen using fast Fourier transform,” Opt. Eng. 53, 016110 (2014). [CrossRef]  

29. G. Sedmak, “Implementation of fast-Fourier-transform-based simulations of extra-large atmospheric phase and scintillation screens,” Appl. Opt. 43, 4527–4538 (2004). [CrossRef]  

30. D. H. Bailey and P. N. Swarztrauber, “The fractional Fourier transform and applications,” SIAM Rev. 33, 389–404 (1995). [CrossRef]  

31. E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994). [CrossRef]  

32. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998). [CrossRef]  

33. R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995). [CrossRef]  

References

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  1. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
    [Crossref]
  2. M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64, 012306 (2001).
    [Crossref]
  3. A. Mair, G. W. A. Vaziri, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [Crossref]
  4. D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre-Gauss modes using the Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
    [Crossref]
  5. J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
    [Crossref]
  6. 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, 5448–5456 (2004).
    [Crossref]
  7. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
    [Crossref]
  8. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–230 (2008).
    [Crossref]
  9. W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17, 17829–17836 (2009).
    [Crossref]
  10. A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E 80, 046609 (2009).
    [Crossref]
  11. N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91, 012345 (2015).
    [Crossref]
  12. A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
    [Crossref]
  13. A. H. Ibrahim, F. S. Roux, and T. Konrad, “Parameter dependence in the atmospheric decoherence of modally entangled photon pairs,” Phys. Rev. A 90, 052115 (2014).
    [Crossref]
  14. F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
    [Crossref]
  15. J. R. G. Alonso and T. A. Brun, “Protecting orbital-angular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88, 022326 (2013).
    [Crossref]
  16. B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
    [Crossref]
  17. F. S. Roux, “The Lindblad equation for the decay of entanglement due to atmospheric scintillation,” J. Phys. A 47, 195302 (2014).
    [Crossref]
  18. T. Brűnner and F. S. Roux, “Robust entangled qutrit states in atmospheric turbulence,” New J. Phys. 15, 063005 (2013).
    [Crossref]
  19. M. V. Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled two-photon beams,” Phys. Rev. A 88, 053836 (2013).
    [Crossref]
  20. B.-J. Pros, C. H. Monken, E. R. Elie, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express 19, 6671–6683 (2011).
    [Crossref]
  21. D. Dayton, B. Pierson, and B. Spielbusch, “Atmospheric structure function measurements with a Shack-Hartmann wave front sensor,” Opt. Lett. 17, 1737–1739 (1992).
    [Crossref]
  22. M. S. Belenkii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
    [Crossref]
  23. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
    [Crossref]
  24. D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
    [Crossref]
  25. H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial Gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28, 1016–1021 (2011).
    [Crossref]
  26. P. Zhou, Y. Ma, X. Wang, H. Zhao, and Z. Liu, “Average spreading of a Gaussian beam array in non-Kolmogorov turbulence,” Opt. Lett. 35, 1043–1045 (2010).
    [Crossref]
  27. J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
    [Crossref]
  28. J. S. Xiang, “Fast and accurate simulation of the turbulent phase screen using fast Fourier transform,” Opt. Eng. 53, 016110 (2014).
    [Crossref]
  29. G. Sedmak, “Implementation of fast-Fourier-transform-based simulations of extra-large atmospheric phase and scintillation screens,” Appl. Opt. 43, 4527–4538 (2004).
    [Crossref]
  30. D. H. Bailey and P. N. Swarztrauber, “The fractional Fourier transform and applications,” SIAM Rev. 33, 389–404 (1995).
    [Crossref]
  31. E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
    [Crossref]
  32. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
    [Crossref]
  33. R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).
    [Crossref]

2015 (1)

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91, 012345 (2015).
[Crossref]

2014 (3)

A. H. Ibrahim, F. S. Roux, and T. Konrad, “Parameter dependence in the atmospheric decoherence of modally entangled photon pairs,” Phys. Rev. A 90, 052115 (2014).
[Crossref]

F. S. Roux, “The Lindblad equation for the decay of entanglement due to atmospheric scintillation,” J. Phys. A 47, 195302 (2014).
[Crossref]

J. S. Xiang, “Fast and accurate simulation of the turbulent phase screen using fast Fourier transform,” Opt. Eng. 53, 016110 (2014).
[Crossref]

2013 (4)

J. R. G. Alonso and T. A. Brun, “Protecting orbital-angular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88, 022326 (2013).
[Crossref]

T. Brűnner and F. S. Roux, “Robust entangled qutrit states in atmospheric turbulence,” New J. Phys. 15, 063005 (2013).
[Crossref]

M. V. Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled two-photon beams,” Phys. Rev. A 88, 053836 (2013).
[Crossref]

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

2011 (3)

2010 (1)

2009 (2)

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

A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E 80, 046609 (2009).
[Crossref]

2008 (3)

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–230 (2008).
[Crossref]

D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre-Gauss modes using the Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
[Crossref]

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[Crossref]

2006 (1)

B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[Crossref]

2005 (1)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[Crossref]

2004 (2)

2002 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
[Crossref]

2001 (2)

M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64, 012306 (2001).
[Crossref]

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

1998 (1)

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
[Crossref]

1997 (1)

M. S. Belenkii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[Crossref]

1995 (3)

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

D. H. Bailey and P. N. Swarztrauber, “The fractional Fourier transform and applications,” SIAM Rev. 33, 389–404 (1995).
[Crossref]

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).
[Crossref]

1994 (2)

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
[Crossref]

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

1992 (1)

1988 (1)

Aiello, A.

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[Crossref]

Alonso, J. R. G.

J. R. G. Alonso and T. A. Brun, “Protecting orbital-angular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88, 022326 (2013).
[Crossref]

Bailey, D. H.

D. H. Bailey and P. N. Swarztrauber, “The fractional Fourier transform and applications,” SIAM Rev. 33, 389–404 (1995).
[Crossref]

Barnett, S.

Beland, R. R.

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).
[Crossref]

Belenkii, M. S.

M. S. Belenkii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[Crossref]

Bishop, K. P.

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

Björk, G.

M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64, 012306 (2001).
[Crossref]

Bourennane, M.

M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64, 012306 (2001).
[Crossref]

Brown, J. M.

M. S. Belenkii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[Crossref]

Brun, T. A.

J. R. G. Alonso and T. A. Brun, “Protecting orbital-angular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88, 022326 (2013).
[Crossref]

Brunner, T.

T. Brűnner and F. S. Roux, “Robust entangled qutrit states in atmospheric turbulence,” New J. Phys. 15, 063005 (2013).
[Crossref]

Buchleitner, A.

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91, 012345 (2015).
[Crossref]

Cheng, W.

Courtial, J.

Cunha Pereira, M. V.

M. V. Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled two-photon beams,” Phys. Rev. A 88, 053836 (2013).
[Crossref]

Dang, A.

Dayton, D.

Dipankar, A.

A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E 80, 046609 (2009).
[Crossref]

Elie, E. R.

Eliel, E. R.

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[Crossref]

Filpi, L. A. P.

M. V. Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled two-photon beams,” Phys. Rev. A 88, 053836 (2013).
[Crossref]

Flatté, S. M.

Forbes, A.

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

Franke-Arnold, S.

Fugate, R. Q.

M. S. Belenkii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[Crossref]

Gavel, D. T.

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
[Crossref]

Gbur, G.

Gibson, G.

Guo, H.

Haus, J. W.

Hooft, G. W.

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[Crossref]

Ibrahim, A. H.

A. H. Ibrahim, F. S. Roux, and T. Konrad, “Parameter dependence in the atmospheric decoherence of modally entangled photon pairs,” Phys. Rev. A 90, 052115 (2014).
[Crossref]

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

Johansson, E. M.

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
[Crossref]

Karis, S. J.

M. S. Belenkii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[Crossref]

Karlsson, A.

M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64, 012306 (2001).
[Crossref]

Kawase, D.

D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre-Gauss modes using the Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
[Crossref]

Keating, D. D. B.

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

Konrad, T.

A. H. Ibrahim, F. S. Roux, and T. Konrad, “Parameter dependence in the atmospheric decoherence of modally entangled photon pairs,” Phys. Rev. A 90, 052115 (2014).
[Crossref]

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

Kyrazis, D. T.

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

Leonhard, N. D.

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91, 012345 (2015).
[Crossref]

Liu, Z.

Luo, B.

Ma, Y.

Mair, A.

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

Marchiano, R.

A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E 80, 046609 (2009).
[Crossref]

Martin, J. M.

McLaren, M.

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

Miyamoto, Y.

D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre-Gauss modes using the Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
[Crossref]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
[Crossref]

Monken, C. H.

M. V. Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled two-photon beams,” Phys. Rev. A 88, 053836 (2013).
[Crossref]

B.-J. Pros, C. H. Monken, E. R. Elie, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express 19, 6671–6683 (2011).
[Crossref]

Oemrawsingh, S. S. R.

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[Crossref]

Ou, B.

Padgett, M.

Pas’ko, V.

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[Crossref]

Pierson, B.

Pors, J. B.

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[Crossref]

Preble, A. J.

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

Pros, B.-J.

Raymer, M. G.

B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[Crossref]

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Roux, F. S.

F. S. Roux, “The Lindblad equation for the decay of entanglement due to atmospheric scintillation,” J. Phys. A 47, 195302 (2014).
[Crossref]

A. H. Ibrahim, F. S. Roux, and T. Konrad, “Parameter dependence in the atmospheric decoherence of modally entangled photon pairs,” Phys. Rev. A 90, 052115 (2014).
[Crossref]

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

T. Brűnner and F. S. Roux, “Robust entangled qutrit states in atmospheric turbulence,” New J. Phys. 15, 063005 (2013).
[Crossref]

F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
[Crossref]

Sagaut, P.

A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E 80, 046609 (2009).
[Crossref]

Sasaki, K.

D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre-Gauss modes using the Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
[Crossref]

Sedmak, G.

Shatokhin, V. N.

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91, 012345 (2015).
[Crossref]

Smith, B. J.

B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[Crossref]

Spielbusch, B.

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Swarztrauber, P. N.

D. H. Bailey and P. N. Swarztrauber, “The fractional Fourier transform and applications,” SIAM Rev. 33, 389–404 (1995).
[Crossref]

Takeda, M.

D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre-Gauss modes using the Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
[Crossref]

Takeuchi, S.

D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre-Gauss modes using the Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
[Crossref]

Tang, H.

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
[Crossref]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
[Crossref]

Tyson, R. K.

van Exter, M. P.

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[Crossref]

Vasnetsov, M.

Vaziri, G. W. A.

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

Wang, X.

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Wissler, J.

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

Woerdman, J. P.

B.-J. Pros, C. H. Monken, E. R. Elie, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express 19, 6671–6683 (2011).
[Crossref]

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[Crossref]

Wootters, W. K.

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
[Crossref]

Xiang, J. S.

J. S. Xiang, “Fast and accurate simulation of the turbulent phase screen using fast Fourier transform,” Opt. Eng. 53, 016110 (2014).
[Crossref]

Zeilinger, A.

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

Zhan, Q.

Zhao, H.

Zhou, P.

Appl. Opt. (2)

J. Opt. Soc. Am. A (2)

J. Phys. A (1)

F. S. Roux, “The Lindblad equation for the decay of entanglement due to atmospheric scintillation,” J. Phys. A 47, 195302 (2014).
[Crossref]

Nature (1)

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

New J. Phys. (1)

T. Brűnner and F. S. Roux, “Robust entangled qutrit states in atmospheric turbulence,” New J. Phys. 15, 063005 (2013).
[Crossref]

Opt. Eng. (1)

J. S. Xiang, “Fast and accurate simulation of the turbulent phase screen using fast Fourier transform,” Opt. Eng. 53, 016110 (2014).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. A (8)

M. V. Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled two-photon beams,” Phys. Rev. A 88, 053836 (2013).
[Crossref]

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91, 012345 (2015).
[Crossref]

A. H. Ibrahim, F. S. Roux, M. McLaren, T. Konrad, and A. Forbes, “Orbital-angular-momentum entanglement in turbulence,” Phys. Rev. A 88, 012312 (2013).
[Crossref]

A. H. Ibrahim, F. S. Roux, and T. Konrad, “Parameter dependence in the atmospheric decoherence of modally entangled photon pairs,” Phys. Rev. A 90, 052115 (2014).
[Crossref]

F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
[Crossref]

J. R. G. Alonso and T. A. Brun, “Protecting orbital-angular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88, 022326 (2013).
[Crossref]

B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[Crossref]

M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64, 012306 (2001).
[Crossref]

Phys. Rev. E (1)

A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E 80, 046609 (2009).
[Crossref]

Phys. Rev. Lett. (5)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[Crossref]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002).
[Crossref]

D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre-Gauss modes using the Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
[Crossref]

J. B. Pors, S. S. R. Oemrawsingh, A. Aiello, M. P. van Exter, E. R. Eliel, G. W. Hooft, and J. P. Woerdman, “Shannon dimensionality of quantum channels and its application to photon entanglement,” Phys. Rev. Lett. 101, 120502 (2008).
[Crossref]

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
[Crossref]

Proc. SPIE (5)

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).
[Crossref]

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
[Crossref]

M. S. Belenkii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[Crossref]

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

D. T. Kyrazis, J. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[Crossref]

SIAM Rev. (1)

D. H. Bailey and P. N. Swarztrauber, “The fractional Fourier transform and applications,” SIAM Rev. 33, 389–404 (1995).
[Crossref]

Cited By

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Figures (7)

Fig. 1.
Fig. 1. Source produces a pairs of OAM-entangled qubits, whose wave fronts get deteriorated as they propagate through a series of phase screens along the (horizontal) z axis toward a detector.
Fig. 2.
Fig. 2. Comparisons between Eq. (11) and statistical mean of the structure function obtained from samples of 1000 phase screens generated with the spectral method. The black color is the results of Eq. (11), while the red represents the sample average results. The value of the Fried parameter is r0=3.18  cm.
Fig. 3.
Fig. 3. Concurrence as a function of ratio ω0/ρ0 when both the photons propagate through turbulence. In (a) l=1, in (b) l=3, in (c) l=5, and in (d) l=7.
Fig. 4.
Fig. 4. Concurrence as a function of ratio ω0/ρ0 for different values of the azimuthal modes.
Fig. 5.
Fig. 5. Concurrence as a function of ratio ω0/ρ0 for different values of the generalized exponent.
Fig. 6.
Fig. 6. Concurrence as a function of ratio ω0/ρ0 for different values of the turbulence outer scale with α=3.6.
Fig. 7.
Fig. 7. Concurrence as a function of ratio ω0/ρ0 for different values of the turbulence inner scale with α=3.6.

Equations (12)

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

|ψin=12(|A|B+|A|B),
MpLG=Nr||exp(iφ)(1+it)p(1it)p+||+1LP||(2r21+t2)exp(r21it),
N=(p!2||+1π(p+||)!)1/2.
δn(x)=n1,
φ(x)=k00Δzδn(x)dz,
Φn(κ,z)=A(α)C˜n2(κ2+κ02)α/2exp(κ2/κm2),3<α<4,
A(α)=Γ(α1)cos(απ/2)/(4π2),
Φn(κ)=0.033Cn2κ11/3.
φFFT(m,n)=m=N/2N/21n=N/2N/21h(m,n)Δκ2πk2Φn(κx,κy)Δzei(mm/N+nn/N)ΔκΔκ,
φLow(m,n)=m=Nx/2Nx/21n=Ny/2Ny/21h(m,n)Δκ2πk2Φn(κx,κy)Δzei(mm/NxN+nn/NyN)ΔκΔκ.
Dφ(r)=6.16r05/3[35(L02π)5/3(rL0/4π)5/6Γ(11/6)K5/6(2πrL0)].
ρ0=(A(α)B(α)C˜n2k2L)1α2,

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