Abstract

Mutually orthogonal polarization photons from Type-II parametric down conversion process in a circular microcavity were discussed based on the nonlinear interaction among whispering gallery modes. Furthermore, an onchip polarization entangled photons source can be obtained with integrated pump laser in the same microcavity and polarization beam splitter. Such an entangled photons resource will be useful in many areas, such as microcavity-QED and onchip quantum information techniques.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  26. D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Opt. Lett. 22, 1553–1555 (1997).
    [CrossRef]
  27. S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6, 145–177 (2012).
    [CrossRef]
  28. C. L. Zou, F. W. Sun, C. H. Dong, X. F. Ren, J. M. Cui, X. D. Chen, Z. F. Han, and G. C. Guo, “Broadband integrated polarization beam splitter with surface plasmon,” Opt. Lett. 36, 3630–3632 (2011).
    [CrossRef]
  29. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).
  30. T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74, 051802 (2006).
    [CrossRef]
  31. S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26–28 (2001).
    [CrossRef]

2012 (1)

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6, 145–177 (2012).
[CrossRef]

2011 (3)

C. L. Zou, F. W. Sun, C. H. Dong, X. F. Ren, J. M. Cui, X. D. Chen, Z. F. Han, and G. C. Guo, “Broadband integrated polarization beam splitter with surface plasmon,” Opt. Lett. 36, 3630–3632 (2011).
[CrossRef]

X. S. Ma, B. Dakic, W. Naylor, A. Zeilinger, and P. Walther, “Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems,” Nat. Phys. 7, 399–405 (2011).
[CrossRef]

T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett. 106, 143903 (2011).
[CrossRef]

2010 (6)

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett. 105, 263904 (2010).
[CrossRef]

L. M. Duan and C. Monroe, “Colloquium: quantum networks with trapped ions,” Rev. Mod. Phys. 82, 1209–1224 (2010).
[CrossRef]

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sorensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010).
[CrossRef]

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nat. Photon. 4, 302–306 (2010).
[CrossRef]

A. J. Bennett, M. A. Pooley, R. M. Stevenson, M. B. Ward, R. B. Patel, A. Boyer de la Giroday, N. Sköld, I. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J. Shields, “Electric-field-induced coherent coupling of the exciton states in a single quantum dot,” Nat. Phys. 6, 947–950 (2010).
[CrossRef]

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
[CrossRef]

2009 (1)

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photon. 3, 687–695 (2009).
[CrossRef]

2008 (5)

X. H. Bao, Y. Qian, J. Yang, H. Zhang, Z. B. Chen, T. Yang, and J. W. Pan, “Generation of narrow-band polarization-entangled photon pairs for atomic quantum memories,” Phys. Rev. Lett. 101, 190501 (2008).
[CrossRef]

L. X. He, M. Gong, C. F. Li, G. C. Guo, and A. Zunger, “Highly reduced fine-structure splitting in InAs/InP quantum dots offering an efficient on-demand entangled 1.55 μm photon emitter,” Phys. Rev. Lett. 101, 157405 (2008).
[CrossRef]

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[CrossRef]

F. W. Sun, B. H. Liu, Y. X. Gong, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Experimental demonstration of phase measurement precision beating standard quantum limit by projection measurement,” Europhys. Lett. 82, 24001 (2008).
[CrossRef]

G. Kozyreff, J. L. Dominguez Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
[CrossRef]

2007 (1)

T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316, 726–729 (2007).
[CrossRef]

2006 (1)

T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74, 051802 (2006).
[CrossRef]

2005 (2)

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[CrossRef]

X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, “Optical-fiber source of polarization-entangled photons in the 1550 nm telecom band,” Phys. Rev. Lett. 94, 053601 (2005).
[CrossRef]

2004 (1)

L. M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004).
[CrossRef]

2001 (2)

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef]

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26–28 (2001).
[CrossRef]

2000 (1)

O. Benson, C. Santori, M. Pelton, and Y. Yamamoto, “Regulated and entangled photons from a single quantum dot,” Phys. Rev. Lett. 84, 2513–2516 (2000).
[CrossRef]

1999 (1)

Z. Y. Ou, and Y. J. Lu, “Cavity enhanced spontaneous parametric down conversion for the prolongation of correlation time between conjugate photons,” Phys. Rev. Lett. 83, 2556–2559 (1999).
[CrossRef]

1997 (1)

D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Opt. Lett. 22, 1553–1555 (1997).
[CrossRef]

1995 (1)

P. Kwiat, K. Mattle, H. Weinfurter, and A. Zeilinger, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75, 4337–4341 (1995).
[CrossRef]

1991 (1)

S. Schiller and R. L. Byer, “High-resolution spectroscopy of whispering gallery modes in large dielectric spheres,” Opt. Lett. 16, 1138–1140 (1991).
[CrossRef]

Aiello, A.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett. 105, 263904 (2010).
[CrossRef]

Andersen, U. L.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett. 105, 263904 (2010).
[CrossRef]

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
[CrossRef]

Aspelmeyer, M.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature 434, 169–176 (2005).
[CrossRef]

Baldi, P.

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26–28 (2001).
[CrossRef]

Bao, X. H.

X. H. Bao, Y. Qian, J. Yang, H. Zhang, Z. B. Chen, T. Yang, and J. W. Pan, “Generation of narrow-band polarization-entangled photon pairs for atomic quantum memories,” Phys. Rev. Lett. 101, 190501 (2008).
[CrossRef]

Beckmann, T.

T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett. 106, 143903 (2011).
[CrossRef]

Bennett, A. J.

A. J. Bennett, M. A. Pooley, R. M. Stevenson, M. B. Ward, R. B. Patel, A. Boyer de la Giroday, N. Sköld, I. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J. Shields, “Electric-field-induced coherent coupling of the exciton states in a single quantum dot,” Nat. Phys. 6, 947–950 (2010).
[CrossRef]

Bennett, C. H.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India (IEEE, 1984), pp. 175–179.

Benson, O.

O. Benson, C. Santori, M. Pelton, and Y. Yamamoto, “Regulated and entangled photons from a single quantum dot,” Phys. Rev. Lett. 84, 2513–2516 (2000).
[CrossRef]

Boyer de la Giroday, A.

A. J. Bennett, M. A. Pooley, R. M. Stevenson, M. B. Ward, R. B. Patel, A. Boyer de la Giroday, N. Sköld, I. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J. Shields, “Electric-field-induced coherent coupling of the exciton states in a single quantum dot,” Nat. Phys. 6, 947–950 (2010).
[CrossRef]

Brassard, G.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India (IEEE, 1984), pp. 175–179.

Breunig, I.

T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett. 106, 143903 (2011).
[CrossRef]

Buse, K.

T. Beckmann, H. Linnenbank, H. Steigerwald, B. Sturman, D. Haertle, K. Buse, and I. Breunig, “Highly tunable low-threshold optical parametric oscillation in radially poled whispering gallery resonators,” Phys. Rev. Lett. 106, 143903 (2011).
[CrossRef]

Byer, R. L.

S. Schiller and R. L. Byer, “High-resolution spectroscopy of whispering gallery modes in large dielectric spheres,” Opt. Lett. 16, 1138–1140 (1991).
[CrossRef]

Cai, H.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6, 145–177 (2012).
[CrossRef]

Chen, H.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6, 145–177 (2012).
[CrossRef]

Chen, X. D.

C. L. Zou, F. W. Sun, C. H. Dong, X. F. Ren, J. M. Cui, X. D. Chen, Z. F. Han, and G. C. Guo, “Broadband integrated polarization beam splitter with surface plasmon,” Opt. Lett. 36, 3630–3632 (2011).
[CrossRef]

Chen, Z. B.

X. H. Bao, Y. Qian, J. Yang, H. Zhang, Z. B. Chen, T. Yang, and J. W. Pan, “Generation of narrow-band polarization-entangled photon pairs for atomic quantum memories,” Phys. Rev. Lett. 101, 190501 (2008).
[CrossRef]

Childress, L.

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sorensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010).
[CrossRef]

Chu, Y.

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sorensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010).
[CrossRef]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Cirac, J. I.

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef]

Cryan, M. J.

A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
[CrossRef]

Cui, J. M.

C. L. Zou, F. W. Sun, C. H. Dong, X. F. Ren, J. M. Cui, X. D. Chen, Z. F. Han, and G. C. Guo, “Broadband integrated polarization beam splitter with surface plasmon,” Opt. Lett. 36, 3630–3632 (2011).
[CrossRef]

Dakic, B.

X. S. Ma, B. Dakic, W. Naylor, A. Zeilinger, and P. Walther, “Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems,” Nat. Phys. 7, 399–405 (2011).
[CrossRef]

De Micheli, M.

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26–28 (2001).
[CrossRef]

De Riedmatten, H.

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26–28 (2001).
[CrossRef]

Dominguez Juarez, J. L.

G. Kozyreff, J. L. Dominguez Juarez, and J. Martorell, “Whispering-gallery-mode phase matching for surface second-order nonlinear optical processes in spherical microresonators,” Phys. Rev. A 77, 043817 (2008).
[CrossRef]

Dong, C. H.

C. L. Zou, F. W. Sun, C. H. Dong, X. F. Ren, J. M. Cui, X. D. Chen, Z. F. Han, and G. C. Guo, “Broadband integrated polarization beam splitter with surface plasmon,” Opt. Lett. 36, 3630–3632 (2011).
[CrossRef]

Duan, L. M.

L. M. Duan and C. Monroe, “Colloquium: quantum networks with trapped ions,” Rev. Mod. Phys. 82, 1209–1224 (2010).
[CrossRef]

L. M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004).
[CrossRef]

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413–418 (2001).
[CrossRef]

Dutt, M. V. G.

E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sorensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010).
[CrossRef]

Dwir, B.

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nat. Photon. 4, 302–306 (2010).
[CrossRef]

Elser, D.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
[CrossRef]

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett. 105, 263904 (2010).
[CrossRef]

Faist, J.

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nat. Photon. 4, 302–306 (2010).
[CrossRef]

Farrer, I.

A. J. Bennett, M. A. Pooley, R. M. Stevenson, M. B. Ward, R. B. Patel, A. Boyer de la Giroday, N. Sköld, I. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J. Shields, “Electric-field-induced coherent coupling of the exciton states in a single quantum dot,” Nat. Phys. 6, 947–950 (2010).
[CrossRef]

Felici, M.

A. Mohan, M. Felici, P. Gallo, B. Dwir, A. Rudra, J. Faist, and E. Kapon, “Polarization-entangled photons produced with high-symmetry site-controlled quantum dots,” Nat. Photon. 4, 302–306 (2010).
[CrossRef]

Feng, S.

S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, and A. W. Poon, “Silicon photonics: from a microresonator perspective,” Laser Photonics Rev. 6, 145–177 (2012).
[CrossRef]

Fürst, J. U.

J. U. Fürst, D. V. Strekalov, D. Elser, A. Aiello, U. L. Andersen, C. Marquardt, and G. Leuchs, “Low-threshold optical parametric oscillations in a whispering gallery mode resonator,” Phys. Rev. Lett. 105, 263904 (2010).
[CrossRef]

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104, 153901 (2010).
[CrossRef]

Furusawa, A.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photon. 3, 687–695 (2009).
[CrossRef]

Gallo, P.

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A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. L. O’Brien, “Silica-on-silicon waveguide quantum circuits,” Science 320, 646–649 (2008).
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S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26–28 (2001).
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X. H. Bao, Y. Qian, J. Yang, H. Zhang, Z. B. Chen, T. Yang, and J. W. Pan, “Generation of narrow-band polarization-entangled photon pairs for atomic quantum memories,” Phys. Rev. Lett. 101, 190501 (2008).
[CrossRef]

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Electron. Lett. (1)

S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi, M. De Micheli, D. B. Ostrowsky, and N. Gisin, “Highly efficient photon-pair source using periodically poled lithium niobate waveguide,” Electron. Lett. 37, 26–28 (2001).
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A. J. Bennett, M. A. Pooley, R. M. Stevenson, M. B. Ward, R. B. Patel, A. Boyer de la Giroday, N. Sköld, I. Farrer, C. A. Nicoll, D. A. Ritchie, and A. J. Shields, “Electric-field-induced coherent coupling of the exciton states in a single quantum dot,” Nat. Phys. 6, 947–950 (2010).
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Nature (3)

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Phys. Rev. A (2)

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Figures (7)

Fig. 1.
Fig. 1.

Normalized radial coupling coefficients σr of different signal and idler modes. The pump light is in the TE mode, and the radial number of pump light is Q=1.

Fig. 2.
Fig. 2.

Contour map of Fig. 1. Valleys with color of dark blue are shown in the figure.

Fig. 3.
Fig. 3.

Normalized angular coupling coefficients σθ,φ of different signal and idler modes. The pump light is equatorial, satisfying L=M=1600, and ms=mi=800.

Fig. 4.
Fig. 4.

ΔL is a function of radius r, with l, m1, Q is small, and li=mi=ls=ms, L=M. When radius r is about 100 μm, the phase-matching condition is attained with the radial numbers qs=qi=1, and Q=2.

Fig. 5.
Fig. 5.

Δkkpkiks is a function of orbital numbers L with different radial numbers. The PDC process occurs when Δk is zero, and from this figure several orbital numbers L can be attained to satisfy this condition. Possible pump wavelengths accordingly are about 537, 573, 632, 656, 793, and 775 nm.

Fig. 6.
Fig. 6.

η are relative conversion efficiencies of different pump wavelengths compared with that of pump light with 775 nm.

Fig. 7.
Fig. 7.

Schematic illustration of the setup to generate an onchip polarization entangled photons source. The microdisk with integrated pump laser is used to prepare two-photon states, which can be transformed to polarization entangled photons states by the coupling waveguide and the onchip PBS [28].

Equations (7)

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

H=χ(2)Ep*EsEidV,
E=E0Ylm(θ,φ)jl(klqr),E0={l(l+1)kr,TM modemsinθ,TE mode,
klq=1r[(l+12)21/3ζq(l+12)1/3β(n21)1/2],
H=χ(2)σrσθ,φ,
σθ,φ=YlsmsYlimiYLM*sinθdθdφ,
σr=EjlsjlijLr2dr,
M=mi+ms,|lils|L|li+ls|,L+li+lseven integer.

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