## Flux enhancement of photons entangled in orbital angular momentum |

Optics Express, Vol. 19, Issue 15, pp. 14108-14120 (2011)

http://dx.doi.org/10.1364/OE.19.014108

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### Abstract

Entangled photons are generally collected by detection systems that select their certain spatial modes, for example using single-mode optical fibers. We derive *simple* and *easy-to-use* expressions that allow us to maximize the coupling efficiency of entangled photons with specific orbital angular momentum (OAM) correlations generated by means of spontaneous parametric downconversion. Two different configurations are considered: one in which the beams with OAM are generated by conversion from beams without OAM, and the second when beams with OAM are generated directly from the nonlinear medium. Also, an example of how to generate a maximally entangled qutrit is presented.

© 2011 OSA

## 1. Introduction

5. B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. **94**, 043602 (2005). [CrossRef] [PubMed]

*μ*W), that even while being orders of magnitude greater than what is typically utilized in quantum optics experiments, it still shows an efficiency of only ∼ 10

^{−7}. In order to increase the flux of paired photons, one has to choose longer nonlinear materials or materials with higher nonlinear coefficients.

6. C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “High-efficiency entangled photon pair collection in type-II parametric fluorescence,” Phys. Rev. A **64**, 023802 (2001). [CrossRef]

10. R. S. Bennink, “Optimal collinear Gaussian beams for spontaneous parametric down-conversion,” Phys. Rev. A **81**, 053805 (2010). [CrossRef]

11. A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Opt. Express **15**, 15377–15386 (2007). [CrossRef] [PubMed]

12. J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A **68**, 050301 (2003). [CrossRef]

13. 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**, 020505 (2010). [CrossRef] [PubMed]

*spin-orbital*coupling devices that starting from polarization entangled photons with a Gaussian spatial shape, generate photon pairs with OAM and polarization correlations [14

14. E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. **103**, 013601 (2009). [CrossRef] [PubMed]

15. J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A **67**, 052313 (2003). [CrossRef]

## 2. General equations

*L*and nonlinear coefficient

*χ*

^{(2)}illuminated by a continuous wave (CW) pump beam, with central frequency

*ω*. All the interacting waves (the pump, signal and idler) propagate along the same direction (

_{p}*collinear configuration*), and the signal-idler pairs can be distinguished because either they show orthogonal polarizations or because they have different central frequencies. We consider a non-critical configuration, i.e., neither of the interacting waves experiences a Poynting-vector walk-off. The absence of spatial walk-off allows one to employ longer nonlinear media. Narrowband filters are located in front of the single-photon counting modules that detect in coincidence the arrival of a pair of photons.

*m*per photon.

_{p}h̄**q**= (

*q*,

_{x}*q*) is the transverse wavevector and

_{y}*w*is the beam waist.

_{p}*E*

_{0}is a normalizing constant, so that ∫

*d*

**q**|

*E*(

_{p}**q**)|

^{2}= 1.

*U*) allows one to describe the spatial quantum correlations of the paired photons in a straightforward and clear way. The Laguerre-Gaussian modes are characterized by two integer indices,

_{m,p}*p*and

*m*. The positive index

*p*is the radial index, and the winding number

*m*, which can be any integer number, determines the azimuthal phase dependence of the mode, which is of the form ∼ exp(

*imφ*). The functions

*U*are normalized, i.e., ∫

_{m,p}*d*

**q**|

*U*(

_{mp}**q**)|

^{2}= 1.

*m*=

_{p}*m*

_{1}+

*m*

_{2}. Here

*m*is the optical vortex winding number of the pump beam, and

_{p}*m*

_{1}and

*m*

_{2}are the winding numbers of the modes into which the quantum states of the signal and idler photons are projected, respectively. The configuration considered here allows to establish clear spatial correlations between the photons in terms of modes with OAM. For instance, if the pump beam has a Gaussian spatial shape (

*m*= 0), only paired photons with

_{p}*m*

_{1}= −

*m*

_{2}can be detected in coincidence. It is important to remark that this might not be the case for other non-collinear or critical SPDC configurations [16

16. C. I. Osorio, G. Molina-Terriza, and J. P. Torres, “Correlations in orbital angular momentum of spatially entangled paired photons generated in parametric down-conversion,” Phys. Rev. A **77**, 015810 (2008). [CrossRef]

*a*

_{1}and

*a*

_{2}, that evolve as a function of the propagation distance inside the nonlinear medium. In the Heisenberg picture,

*all*the coherence functions of interest can be easily calculated, even when a first-order approximation of the quantum evolution equations is used, as it is the case here [17].

18. A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A **68**, 053807 (2003). [CrossRef]

**q**

_{1}and

**q**

_{2}are the transverse wavenumbers of the signal and idler photons, respectively, and

*ω*

_{1}=

*ω*

_{2}=

*ω*/2 are the central frequencies.

_{p}*n*(

_{i}*i*=

*p*, 1, 2) are the refractive index at the corresponding central frequency,

*F*is the total flux (photons/s) of pump photons that traverse the nonlinear crystal.

_{p}**q**

_{0}– Δ

**q**/2 and

**q**

_{0}+ Δ

**q**/2. Therefore, the overall spectral brightness is We are interested in calculating the spectral brightness

*F*

^{m1,p1,m2,p2}of photons that are detected in coincidence, when the signal photon is projected into a spatial mode with index (

*m*

_{1},

*p*

_{1}) and the idler photon is projected into a spatial mode with index (

*m*

_{2},

*p*

_{2}). The nature and strength of the correlations between paired photons projected into specific spatial modes is determined by the second-order correlation function between signal and idler photons where

*b*

_{m1,p1}and

*c*

_{m2,p2}are operators acting on the signal and idler photons, respectively, that write

*m*

_{1},

*p*

_{1}) and (

*m*

_{2},

*p*

_{2}), respectively, and

19. W. Wasilewski, A. I. Lovsky, K. Banaszek, and C. Radzewicz, “Pulse squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A **73**, 063819 (2006). [CrossRef]

*=*

_{k}*k*(

_{p}**q**

_{1}+

**q**

_{2}) −

*k*

_{1}(

**q**

_{1}) −

*k*

_{2}(

**q**

_{2}) −

*K*is the phase-matching function,

_{g}*K*= 2

_{g}*π*/Λ is the grating vector of the periodically-poled crystal and Λ is the period of the nonlinear grating.

*i*=

*p,*1

*,*2), where

*n*≅

_{p}*n*

_{1}≅

*n*

_{2}, we obtain [20

20. S. P. Walborn, A. N. de Oliveira, S. Padua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. **90**, 143601 (2003). [CrossRef] [PubMed]

*F*can be calculated Additionally, using Eq. (12), we obtain

*F*depends on the total flux of pump photons

*F*that traverse the nonlinear crystal, but not on the specific spatial shape of the pump beam. Notice also that it increases linearly with the length of the crystal.

_{p}*U*, in what follows we will restrict ourselves to the case

_{m,p}*p*

_{1}=

*p*

_{2}= 0. The spatial shape of the LG modes in which the signal and idler photons will be projected,

*U*≡

_{m}*U*

_{m,p}_{=0}writes In order to simplify the notation, we will designate

## 3. Generation of two-photon entangled states with correlations in polarization and OAM

14. E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. **103**, 013601 (2009). [CrossRef] [PubMed]

*R*and

*L*represent right-handed and left-handed circular polarizations, respectively. The procedure to generate such a state consists of two steps. First, a polarization entangled state embedded in Gaussian modes is generated. And second, properly designed diffractive elements are used to transform the polarization and spatial shape of the photons. For instance, one can use a diffractive element that performs the following transformation [21

21. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. **96**, 163905 (2006). [CrossRef] [PubMed]

22. T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” Phys. Rev. A **73**, 012316 (2006). [CrossRef]

*P*

_{0,0}. It reaches its maximum value when the conditions are fulfilled. The non-zero value of

*α*that fulfills Eq. (23) is easily found to be

*α*= 1.39. This solution gives us the optimum value of the pump beam waist, Equations (22) and (24) give us the optimum values of the pump beam waist and the collection system that maximizes the number of generated photons with the desired spatial shape.

*w̄*, that yields the maximum value of coupling efficiency

_{s}*P*

_{0,0}for a given value of the pump beam waist

*w*. Figure 2(b) shows the maximum value of

_{p}*P*

_{0,0}for each value of the pump beam waist. We see that for all lengths, the global maximum of

*P*

_{0,0}is

*L*= 10 mm and

*L*= 20 mm.

*global*maximum of

*P*

_{0,0}is obtained for

*w̄*), no longer fulfills the condition

_{s}*w*>

_{p}*w̄*is indeed smaller than

_{s}*w*<

_{p}*w̄*is larger than

_{s}*w̄*yields a local maximum of the coupling efficiency for a fixed value of

_{s}*w*. To highlight the different coupling efficiency achieved when

_{p}*w*=

_{s}*w̄*and when

_{s}*P*

_{0,0}that can be achieved with a pump beam waist given by Eq. (24) and collection system

24. P. J. Winzer and W. R. Leeb, “Fiber coupling efficiency for random light and its applications to lidar,” Opt. Lett. **23**, 986–988 (1998). [CrossRef]

*η*= 81.5% if the field of the fundamental mode of the fiber is approximated by a Gaussian function. On the other hand, if the exact form of the fundamental mode is used [25

_{max}25. F. Fidler and O. Wallner, “Application of single-mode fiber-coupled receivers in optical satellite to high-altitude platform communications,” EURASIP J. Wireless Commun. Netw. **2008**, 864031 (2008). [CrossRef]

*η*= 78.6%.

_{max}*η*accounts for all the losses of the experimental set-up, i.e., efficiency of the diffractive elements, losses of the singlemode-fibers and quantum efficiency of the detectors, the spectral brightness

*L*= 20 mm PPKTP nonlinear crystal with nonlinear coefficient

*χ*

^{(2)}∼ 10 pm/V, pumped by a CW pump at λ

*= 405 nm and refractive indices*

_{p}*n*≃

_{p}*n*

_{1}≃

*n*

_{2}= 1.8. If the total losses of the optical systems are

*η*= 0.1, the spectral brightness is

*F̄*=

*γP*(

_{p}*mW*), where

*γ*∼ 1 × 10

^{5}photons/s/nm/mW and

*P*is the pump power. Comparing with the flux of downconverted photons measured in [14

_{p}14. E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. **103**, 013601 (2009). [CrossRef] [PubMed]

*w*and the optimum size of the collection mode

_{p}*w*, one could observe a noteworthy enhancement of the spectral brightness of the source.

_{s}## 4. Generation of two-photon entangled states with OAM correlations

*d*= 2. If the goal is to generate entanglement in a multidimensional Hilbert space (

*d >*2), we can take advantage of the OAM correlations directly generated in the process of SPDC (see Fig. 3).

*m*. The two-photon state, which is entangled in the OAM degree of freedom, can be written as where

_{p}*m*=

_{p}*m*

_{1}+

*m*

_{2}. In this section, the goal is to maximize the overall coupling efficiency

*P*= Σ

_{m1,m2}

*P*

_{m1,m2}for a selected set of values of

*m*

_{1}, where

*m*

_{1}and

*m*

_{2}have the same sign, i.e., sgn(

*m*

_{1})

*·*sgn(

*m*

_{2}) = 1, and secondly, the case with sgn(

*m*

_{1})

*·*sgn(

*m*

_{2}) = −1. The first situation corresponds, for instance, to cases where the pump beam, and the signal and idler modes are described by Laguerre-Gaussian modes with positive indices

*m*,

_{p}*m*

_{1}and

*m*

_{2}. Making use of Eqs. (13) and (15), we obtain Notice that by setting

*m*=

_{p}*m*

_{1}=

*m*

_{2}= 0, we recover Eq. (21).

*m*= 1, the fraction of signal photons with

_{p}*m*

_{1}= 0 detected in coincidence with idler photons with

*m*

_{2}= 1 is

*m*= 2, we obtain

_{p}*m*

_{1}) · sgn(

*m*

_{2}) = −1. For a Gaussian pump beam (

*m*= 0), the coupling efficiency is given by a somehow more cumbersome expression that writes

_{p}*m*

_{1}= 1 and

*m*

_{1}= −1. In this case, Eq. (30) can be written as The qualitative behaviour of

*P*

_{1,−1}turns out to be quite different from the previously considered cases, where

*m*

_{1}and

*m*

_{2}had the same sign. This can be clearly observed in Fig. 4(b), which shows the maximum coupling efficiency that can be achieved as a function of the pump beam waist, when the optimum value of

*w*, given by Fig. 4(a), is chosen. Figure 4(b) also shows the

_{s}*v*optimum value of

*P*

_{0,0}(red dash-dotted line), and

*P*

_{0,0}when setting

*m*

_{1,2}= −1,+1, we can make use of a properly designed hologram of order

*M*= +1,−1, or a spatial light modulator. The hologram projects the incoming photons into outgoing photons (in reflection or transmission) that propagate along the first diffraction order of the hologram, and have a winding number

*m*+

*M*. After the hologram, a single-mode fiber detects photons with a Gaussian shape, i.e., with winding number

*m*+

*M*= 0. For instance, if

*M*= +1, detection of a photon after traversing the single-mode fiber implies the presence of an incoming photon with winding number

*m*= −1, since any other incoming photon with a different winding number would not be allowed to propagate inside the single-mode fiber.

*U*

_{0}and

*U*

_{1}in Eq. (33) can be different to achieve a maximum efficiency. In the detection arrangement described above, the beam waist of the modes detected can be controlled by modifying the optical coupling system of light into the single-mode fiber. The weights of each mode,

*P*

_{0,0}= |

*C*

_{0,0}|

^{2}and

*P*

_{1,−1}=

*P*

_{−1,1}= |

*C*

_{−1,1}|

^{2}can be read in Fig. 4(b). To generate a maximally entangled qutrit, i.e., a quantum state that fulfills the condition

*P*

_{1,−1}=

*P*

_{0,0}, one has to use specific values of

*w*and

_{p}*w*. For instance, from Fig. 4(b), we can see that if we make use of the condition

_{s}*w*∼ 55

_{p}*μ*m where the curves intersect corresponds to a high-flux configuration for a maximally-entangled qutrit with

*P*

_{0,0}=

*P*

_{−1,1}. For this value of

*w*, the quantum state given by Eq. (32) represents roughly a 46% of the whole parameter space, i.e.,

_{p}*w*,

_{p}## 5. Conclusions

*w*) and the waist of the collection mode (

_{p}*w*).

_{s}## Acknowledgments

## References and links

1. | J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. |

2. | E. Nagali, L. Sansoni, L. Marrucci, E. Santamato, and F. Sciarrino, “Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding,” Phys. Rev. A |

3. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. |

4. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature |

5. | B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. |

6. | C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “High-efficiency entangled photon pair collection in type-II parametric fluorescence,” Phys. Rev. A |

7. | F. A. Bovino, P. Varisco, A. M. Colla, G. Castagnoli, G. Di Giuseppe, and A. V. Sergienko, “Effective fiber-coupling of entangled photons for quantum communication,” Opt. Commun. |

8. | M. Pelton, P. Marsden, D. Ljunggren, M. Tengner, A. Karlsson, A. Fragemann, C. Canalias, and F. Laurell, “Bright, single-spatial-mode source of frequency non-degenerate, polarization-entangled photon pairs using periodically poled KTP,” Opt. Express |

9. | D. Ljunggren and M. Tengner, “Optimal focusing for maximal collection of entangled narrow-band photon pairs into single-mode fibers,” Phys. Rev. A |

10. | R. S. Bennink, “Optimal collinear Gaussian beams for spontaneous parametric down-conversion,” Phys. Rev. A |

11. | A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Opt. Express |

12. | J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A |

13. | 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. |

14. | E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. |

15. | J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A |

16. | C. I. Osorio, G. Molina-Terriza, and J. P. Torres, “Correlations in orbital angular momentum of spatially entangled paired photons generated in parametric down-conversion,” Phys. Rev. A |

17. | R. Loudon, |

18. | A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A |

19. | W. Wasilewski, A. I. Lovsky, K. Banaszek, and C. Radzewicz, “Pulse squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A |

20. | S. P. Walborn, A. N. de Oliveira, S. Padua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. |

21. | L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. |

22. | T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” Phys. Rev. A |

23. | W. P. Grice, R. S. Bennink, D. S. Goodman, and A. T. Ryan, “Spatial entanglement and optimal single-mode coupling,” Phys. Rev. A |

24. | P. J. Winzer and W. R. Leeb, “Fiber coupling efficiency for random light and its applications to lidar,” Opt. Lett. |

25. | F. Fidler and O. Wallner, “Application of single-mode fiber-coupled receivers in optical satellite to high-altitude platform communications,” EURASIP J. Wireless Commun. Netw. |

**OCIS Codes**

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: May 10, 2011

Revised Manuscript: June 20, 2011

Manuscript Accepted: June 20, 2011

Published: July 8, 2011

**Citation**

Silvana Palacios, R. de J. León-Montiel, Martin Hendrych, Alejandra Valencia, and Juan P. Torres, "Flux enhancement of photons entangled in orbital angular momentum," Opt. Express **19**, 14108-14120 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14108

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### References

- J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett. 95, 260501 (2005). [CrossRef]
- E. Nagali, L. Sansoni, L. Marrucci, E. Santamato, and F. Sciarrino, “Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding,” Phys. Rev. A 81, 052317 (2010). [CrossRef]
- G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007). [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] [PubMed]
- B. Dayan, A. Pe’er, A. A. Friesem, and Y. Silberberg, “Nonlinear interactions with an ultrahigh flux of broadband entangled photons,” Phys. Rev. Lett. 94, 043602 (2005). [CrossRef] [PubMed]
- C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “High-efficiency entangled photon pair collection in type-II parametric fluorescence,” Phys. Rev. A 64, 023802 (2001). [CrossRef]
- F. A. Bovino, P. Varisco, A. M. Colla, G. Castagnoli, G. Di Giuseppe, and A. V. Sergienko, “Effective fiber-coupling of entangled photons for quantum communication,” Opt. Commun. 227, 343–348 (2003). [CrossRef]
- M. Pelton, P. Marsden, D. Ljunggren, M. Tengner, A. Karlsson, A. Fragemann, C. Canalias, and F. Laurell, “Bright, single-spatial-mode source of frequency non-degenerate, polarization-entangled photon pairs using periodically poled KTP,” Opt. Express 12, 3573–3580 (2004). [CrossRef] [PubMed]
- D. Ljunggren and M. Tengner, “Optimal focusing for maximal collection of entangled narrow-band photon pairs into single-mode fibers,” Phys. Rev. A 72, 062301 (2005). [CrossRef]
- R. S. Bennink, “Optimal collinear Gaussian beams for spontaneous parametric down-conversion,” Phys. Rev. A 81, 053805 (2010). [CrossRef]
- A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Opt. Express 15, 15377–15386 (2007). [CrossRef] [PubMed]
- J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301 (2003). [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, 020505 (2010). [CrossRef] [PubMed]
- E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Quantum information transfer from spin to orbital angular momentum of photons,” Phys. Rev. Lett. 103, 013601 (2009). [CrossRef] [PubMed]
- J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003). [CrossRef]
- C. I. Osorio, G. Molina-Terriza, and J. P. Torres, “Correlations in orbital angular momentum of spatially entangled paired photons generated in parametric down-conversion,” Phys. Rev. A 77, 015810 (2008). [CrossRef]
- R. Loudon, The Quantum Theory of Light (Oxford University Press, 1973).
- A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003). [CrossRef]
- W. Wasilewski, A. I. Lovsky, K. Banaszek, and C. Radzewicz, “Pulse squeezed light: simultaneous squeezing of multiple modes,” Phys. Rev. A 73, 063819 (2006). [CrossRef]
- S. P. Walborn, A. N. de Oliveira, S. Padua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003). [CrossRef] [PubMed]
- L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006). [CrossRef] [PubMed]
- T. Kim, M. Fiorentino, and F. N. C. Wong, “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” Phys. Rev. A 73, 012316 (2006). [CrossRef]
- W. P. Grice, R. S. Bennink, D. S. Goodman, and A. T. Ryan, “Spatial entanglement and optimal single-mode coupling,” Phys. Rev. A 83, 023810 (2011). [CrossRef]
- P. J. Winzer and W. R. Leeb, “Fiber coupling efficiency for random light and its applications to lidar,” Opt. Lett. 23, 986–988 (1998). [CrossRef]
- F. Fidler and O. Wallner, “Application of single-mode fiber-coupled receivers in optical satellite to high-altitude platform communications,” EURASIP J. Wireless Commun. Netw. 2008, 864031 (2008). [CrossRef]

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