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

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2011 (1)

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]

2010 (3)

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]

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

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

2009 (1)

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]

2008 (2)

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]

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]

2007 (2)

2006 (3)

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]

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]

2005 (3)

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]

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]

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]

2004 (1)

2003 (5)

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]

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]

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

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]

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]

2001 (2)

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]

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]

1998 (1)

1973 (1)

R. Loudon, The Quantum Theory of Light (Oxford University Press, 1973).

Alexandrescu, A.

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

Banaszek, K.

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]

Barreiro, J. T.

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]

Bennink, R. S.

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]

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

Bovino, F. A.

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]

Canalias, C.

Castagnoli, G.

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]

Colla, A. M.

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]

Dayan, B.

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]

De Martini, F.

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]

de Oliveira, A. N.

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]

Deyanova, Y.

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]

Di Giuseppe, G.

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]

Di Lorenzo Pires, H.

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]

Fedrizzi, A.

Fidler, F.

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]

Fiorentino, M.

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]

Florijn, H. C. B.

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]

Fragemann, A.

Friesem, A. A.

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]

Gatti, A.

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]

Goodman, D. S.

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]

Grice, W. P.

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]

Herbst, T.

Jennewein, T.

Karimi, E.

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]

Karlsson, A.

Kim, T.

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]

Kurtsiefer, C.

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]

Kwiat, P. G.

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]

Langford, N. K.

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]

Laurell, F.

Leeb, W. R.

Ljunggren, D.

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford University Press, 1973).

Lovsky, A. I.

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]

Lugiato, L. A.

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]

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] [PubMed]

Manzo, C.

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]

Marrucci, L.

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]

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]

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]

Marsden, P.

Molina-Terriza, G.

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]

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

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]

Monken, C. H.

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]

Nagali, E.

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]

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]

Oberparleiter, M.

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]

Osorio, C. I.

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]

Padua, S.

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]

Paparo, D.

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]

Pe’er, A.

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]

Pelton, M.

Peters, N. A.

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]

Piccirillo, B.

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]

Poppe, A.

Radzewicz, C.

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]

Ryan, A. T.

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]

San Miguel, M.

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]

Sansoni, L.

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]

Santamato, E.

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]

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]

Sciarrino, F.

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]

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]

Sergienko, A. V.

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]

Silberberg, Y.

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]

Tengner, M.

Torner, L.

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

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

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]

Torres, J. P.

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]

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

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]

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

van Exter, M. P.

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]

Varisco, P.

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]

Vaziri, 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] [PubMed]

Walborn, S. P.

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]

Wallner, O.

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]

Wasilewski, W.

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]

Weihs, G.

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Nature (1)

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Figures (4)

Fig. 1
Fig. 1

Scheme of the combination of a type-II SPDC source embedded in a Sagnac interferometer and diffractive elements to generate photons entangled in the polarization and spatial degrees of freedom. SMF: Single-mode fiber; QWP: Quarter-wave plate; HWP: Half-wave plate; PBS: Polarization beam splitter; DF: Diffractive element. The linked dot lines represent the existence of entanglement.

Fig. 2
Fig. 2

(a) Optimum beam waist of the collection mode, s that maximizes the coupling efficiency P 0,0 for a given pump beam waist wp for two different nonlinear crystal lengths (blue dashed line: L = 20 mm ; red dash-dotted line: L = 10 mm). The dotted line represents the condition w s = 2 w p . The open circles designate the values when w p = w p o p t . (b) Maximum coupling efficiency P 0,0 as a function of pump beam waist wp . Blue dashed line: maximum for L = 20 mm when the optimum value of ws shown in Fig. 2(a) is used. Red dash-dotted line: maximum for L = 10 mm when the optimum value of ws in Fig. 2(a) is used. Blue (dotted) and red (dash dot-dotted) lines show the coupling efficiency when the signal beam waist is w s = 2 w p for L = 20 mm and L = 10 mm. respectively. Horizontal line: Maximum value of P 0,0 for any crystal length. The open circles designate the global maxima of P 0,0, when w p = w p o p t .

Fig. 3
Fig. 3

Scheme of a type-II SPDC configuration for generating photons entangled in the OAM degree of freedom. Photons in Laguerre-Gaussian modes are directly produced in the nonlinear crystal and later separated by a polarization beam splitter (PBS). The linked dot lines represent the existence of entanglement.

Fig. 4
Fig. 4

(a) Optimum beam waist of the collection mode ws that maximizes the coupling efficiency P 1,−1 for a given pump beam waist (blue solid line). For the sake of comparison, the optimum beam waist of the collection mode that maximizes the coupling efficiency P 0,0 for a given pump beam waist is also plotted (red dash-dotted line). (b) Maximum coupling efficiency P 1,−1 as a function of pump beam waist. Blue solid line: maximum coupling efficiency P 1,−1 when the optimum value of ws , as shown in Fig. 4(a), is used. For the sake of comparison, we also plot the coupling efficiency P 0,0 for two cases. Red dash-dotted line: maximum coupling efficiency P 0,0. Red dash-dot-dotted line: coupling efficiency when the signal beam waist is w s = 2 w p . The nonlinear crystal length is L = 10 mm in all cases.

Equations (34)

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E p ( q ) = E 0 ( q x + i q y ) m p exp ( | q | 2 w p 2 / 4 ) ,
a 1 ( q 1 , z ) z = i σ ˜ d q 2 a 2 ( q 2 , z ) E p ( q 1 + q 2 ) × exp [ i k p ( q 1 + q 2 ) i k 1 ( q 1 ) i k 2 ( q 2 ) ] , a 2 ( q 2 , z ) z = i σ ˜ d q 1 a 1 ( q 1 , z ) E p ( q 1 + q 2 ) × exp [ i k p ( q 1 + q 2 ) i k 1 ( q 1 ) i k 2 ( q 2 ) ] ,
σ ˜ = [ h ¯ ω 1 ω 2 ω p [ χ ( 2 ) ] 2 F p 32 π 2 ɛ 0 c 3 n 1 ( ω 1 ) n 2 ( ω 2 ) n p ( ω p ) ] 1 / 2 .
F = d q a 1 ( q ) a 1 ( q ) .
G 12 ( 2 ) ( m 1 , p 1 , m 2 , p 2 ) = b m 1 , p 1 c m 2 , p 2 c m 2 , p 2 b m 1 , p 1 ,
b m 1 , p 1 = d q U m 1 , p 1 ( q ) a 1 ( q ) , c m 2 , p 2 = d q U m 2 , p 2 ( q ) a 2 ( q ) .
G 12 ( 2 ) ( m 1 , p 1 , m 2 , p 2 ) = F 1 m 1 , p 1 F 2 m 2 , p 2 + F 1 , 2 m 1 , p 1 , m 2 , p 2 ,
F 1 m 1 , p 1 = d q 1 d q 1 U m 1 , p 1 * ( q 1 ) U m 1 , p 1 ( q 1 ) a 1 ( q 1 ) a 1 ( q 1 ) , F 2 m 2 , p 2 = d q 2 d q 2 U m 2 , p 2 * ( q 2 ) U m 2 , p 2 ( q 2 ) a 2 ( q 2 ) a 2 ( q 2 )
f 12 m 1 , p 1 , m 2 , p 2 = d q 1 d q 2 U m 1 , p 1 ( q 1 ) U m 2 , p 2 ( q 2 ) a 1 ( q 1 ) a 2 ( q 2 ) .
a 1 ( q 1 ) a 1 ( q 1 ) = d q 2 Φ * ( q 1 , q 2 ) Φ ( q 1 , q 2 ) , a 1 ( q 1 ) a 2 ( q 2 ) = Φ ( q 1 , q 2 ) .
Φ ( q 1 , q 2 ) = σ ˜ L E p ( q 1 + q 2 ) sinc [ Δ k L 2 ] ,
Δ k | q 1 q 2 | 2 2 k p 0 .
F 12 m 1 , p 1 , m 2 , p 2 = | d q 1 d q 2 Φ ( q 1 , q 2 ) U m 1 * ( q 1 ) U m 2 * ( q 2 ) | 2 .
F = d q 1 d q 2 | Φ ( q 1 , q 2 ) | 2 .
F = π 2 σ 2 F p k p 0 2 L .
F = m 1 , p 1 , m 2 , p 2 F 12 m 1 , p 1 , m 2 , p 2 .
U m ( p ) = ( w s 2 2 π | m | ! ) 1 / 2 ( w s | p | 2 ) | m | exp ( | p | 2 w s 2 / 4 ) exp ( i m φ ) .
| Ψ = 1 2 { | R , m 1 = 1 1 | L , m 1 = 1 2 + | L , m 1 = 1 1 | R , m 1 = 1 2 } ,
| R , m = 0 | L , m = + 1 , | L , m = 0 | R , m = 1 .
| Ψ = 1 2 d q s d q i Φ ( q s , q 1 ) { | H 1 | V 2 + | V 1 | H 2 } .
P 00 = 16 k p 0 w p 2 π L [ w s 2 2 w p 2 + w s 2 arctan ( 2 L k p w s 2 ) ] 2 .
w s o p t = 2 w p o p t
1 2 arctan ( α ) = α 1 + α 2 with α = L k p 0 ( w p opt ) 2
w p o p t = L α k p 0 ,
P 0 , 0 m a x = 4 α π ( tan 1 α ) 2 ,
F ¯ = 0.41 η π 2 σ 2 F p k p 0 L .
| Ψ = m 1 C m 1 , m 2 | m 1 1 | m 2 2 ,
P m 1 , m 2 = 16 k p 0 w s 2 π L m p ! m 1 ! m 2 ! [ w p w s 2 w p 2 + w s 2 ] 2 m p + 2 [ tan 1 ( 2 L k p 0 w s 2 ) ] 2 .
P m 1 , m 2 max = m p ! 8 m p m 1 ! m 2 ! 4 α π ( tan 1 α ) 2 .
P m , m = 4 1 m ( m ! ) 2 k p 0 w p 2 π L [ 2 w s 2 2 w p 2 + w s 2 ] 2 m + 2 | p = 0 m ( 1 ) p [ m ! p ! ( m p ) ! ] 2 × Γ ( m p + 1 ) Γ ( p ) [ ( w s 2 / 8 ) 2 + ( L / 4 k p 0 ) 2 ] p / 2 [ w p 2 4 + w s 2 8 ] p sin { p arctan ( 2 L k p 0 w s 2 ) } | 2 .
P 1 , 1 = 16 k p 0 w p 2 w s 8 π L ( 2 w p 2 + w s 2 ) 4 | arctan ( 2 L k p w s 2 ) 2 k p 0 L ( 2 w p 2 + w s 2 ) 4 L 2 + ( k p 0 w s 2 ) 2 | 2 .
| Ψ = C 0 , 0 | m 1 = 0 1 | m 2 = 0 2 + C 1 , 1 { | m 1 = 1 1 | m 2 = 1 2 + C 1 , 1 | m 1 = 1 1 | m 2 = 1 2 } ,
C 0 , 0 = d q s d q i Φ ( q s , q i ) U 0 ( q s ) U 0 ( q i ) ,
C 1 , 1 = d q s d q i Φ ( q s , q i ) U 1 ( q s ) U 1 ( q i ) .

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