Abstract

A key ingredient in emerging quantum-enhanced technologies is the ability to coherently manipulate and detect superpositions of basis states. In integrated optics implementations, transverse spatial modes supported by multimode structures offer an attractive carrier of quantum superpositions. Here we propose an integrated dynamic mode converter based on the electro-optic effect in nonlinear channel waveguides for deterministic transformations between mutually non-orthogonal bases of spatial modes. We theoretically show its capability to demonstrate a violation of a Bell-type Clauser-Horne-Shimony-Holt inequality by measuring spatially mode-entangled photon pairs generated by an integrated photon pair source. The proposed configuration, numerically studied for the potassium titanyl phosphate (KTP) material, can be easily implemented using standard integrated optical fabrication technology.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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2015 (1)

2014 (2)

M. Jachura, M. Karpiński, C. Radzewicz, and K. Banaszek, “High-visibility nonclassical interference of photon pairs generated in a multimode nonlinear waveguide,” Opt. Express 22, 8624–8632 (2014).
[Crossref] [PubMed]

D. Javůrek, J. Svozilík, and J. Peřina, “Emission of orbital-angular-momentum-entangled photon pairs in a nonlinear ring fiber utilizing spontaneous parametric down-conversion,” Phys. Rev. A 90, 043844 (2014).
[Crossref]

2013 (1)

D. Perry, S. Chadderdon, R. Forber, W. C. Wang, R. Selfridge, and S. Schultz, “Multiaxis electric field sensing using slab coupled optical sensors,” App. Opt. 52, 1968–1977 (2013).
[Crossref]

2012 (3)

M. Karpiński, C. Radzewicz, and K. Banaszek, “Dispersion-based control of modal characteristics for parametric down-conversion in a multimode waveguide,” Opt. Lett. 37, 878–880 (2012).
[Crossref]

M. Karpiński, C. Radzewicz, and K. Banaszek, “Generation of spatially pure photon pairs in a multimode nonlinear waveguide using intermodal dispersion,” Proc. SPIE 8518, 85180J (2012).
[Crossref]

Y. Kang, J. Ko, S. M. Lee, S. K. Choi, B. Y. Kim, and H. S. Park, “Measurement of the entanglement between photonic spatial modes in optical fibers,” Phys. Rev. Lett. 109, 020502 (2012).
[Crossref] [PubMed]

2011 (1)

J. P. Torres, K. Banaszek, and I. A. Walmsley, “Engineering nonlinear optic sources of photonic entanglement,” Prog. Opt. 56, 227–331 (2011).
[Crossref]

2009 (5)

P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett. 103, 233901 (2009).
[Crossref]

M. Karpiński, C. Radzewicz, and K. Banaszek, “Experimental characterization of three-wave mixing in a multi-mode nonlinear KTiOPO4 waveguide,” Appl. Phys. Lett. 94, 181105 (2009).
[Crossref]

A. Martin, V. Cristofori, P. Aboussouam, H. Hermann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “Integrated optical source of polarization entangled photons at 1310 nm,” Opt. Express 17, 1033–1041 (2009).
[Crossref] [PubMed]

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

M. F. Saleh, B. E. A. Saleh, and M. C. Teich, “Modal, spectral, and polarization entanglement in guided-wave parametric down-conversion,” Phys. Rev. A 79, 053842 (2009).
[Crossref]

2008 (1)

2007 (1)

2006 (1)

J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Optimal generation of pulsed entangled photon pairs,” Phys. Rev. A 74, 013802 (2006).
[Crossref]

2004 (1)

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultra-bright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69, 041801 (2004).
[Crossref]

2002 (3)

M. C. Booth, M. Atatüre, G. D. Giuseppe, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Counterpropagating entangled photons from a waveguide with periodic nonlinearity,” Phys. Rev. A 66, 023815 (2002).
[Crossref]

S. Tanzilli, W. Tittel, H. de Riedmatten, H. Zbinden, P. Baldi, M. P. de Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).
[Crossref]

K. Kato and E. Takaoka, “Sellmeier and thermo-optic dispersion formulas for KTP,” App. Opt. 41, 5040–5044 (2002).
[Crossref]

2001 (2)

K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated photons in controlled spatial modes by down conversion in nonlinear waveguides,” Opt. Lett. 26, 1367–1369 (2001).
[Crossref]

K. Sanaka, K. Kawahara, and T. Kuga, “New high-efficiency source of photon pairs for engineering quantum entanglement,” Phys. Rev. Lett. 86, 5620–5623 (2001).
[Crossref] [PubMed]

1999 (2)

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

1998 (1)

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[Crossref]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 74, 4337–4341 (1995).
[Crossref]

1988 (1)

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[Crossref] [PubMed]

1986 (1)

J. D. Bierlein and C. B. Arweiler, “Electro-optic and dielectric properties of KTiOPO4,” Appl. Phys. Lett. 49, 917–919 (1986).
[Crossref]

1975 (1)

D. Marcuse, “Electro optic coupling between TE and TM Modes in Anisotropic Slabs,” IEEE J. Quantum Electron. QE-9, 759–767 (1975).
[Crossref]

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

1969 (1)

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[Crossref]

Aboussouam, P.

Alibart, O.

Appelbaum, I.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

Arweiler, C. B.

J. D. Bierlein and C. B. Arweiler, “Electro-optic and dielectric properties of KTiOPO4,” Appl. Phys. Lett. 49, 917–919 (1986).
[Crossref]

Atatüre, M.

M. C. Booth, M. Atatüre, G. D. Giuseppe, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Counterpropagating entangled photons from a waveguide with periodic nonlinearity,” Phys. Rev. A 66, 023815 (2002).
[Crossref]

Baldi, P.

S. Tanzilli, W. Tittel, H. de Riedmatten, H. Zbinden, P. Baldi, M. P. de Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).
[Crossref]

Banaszek, K.

M. Jachura, M. Karpiński, C. Radzewicz, and K. Banaszek, “High-visibility nonclassical interference of photon pairs generated in a multimode nonlinear waveguide,” Opt. Express 22, 8624–8632 (2014).
[Crossref] [PubMed]

M. Karpiński, C. Radzewicz, and K. Banaszek, “Dispersion-based control of modal characteristics for parametric down-conversion in a multimode waveguide,” Opt. Lett. 37, 878–880 (2012).
[Crossref]

M. Karpiński, C. Radzewicz, and K. Banaszek, “Generation of spatially pure photon pairs in a multimode nonlinear waveguide using intermodal dispersion,” Proc. SPIE 8518, 85180J (2012).
[Crossref]

J. P. Torres, K. Banaszek, and I. A. Walmsley, “Engineering nonlinear optic sources of photonic entanglement,” Prog. Opt. 56, 227–331 (2011).
[Crossref]

M. Karpiński, C. Radzewicz, and K. Banaszek, “Experimental characterization of three-wave mixing in a multi-mode nonlinear KTiOPO4 waveguide,” Appl. Phys. Lett. 94, 181105 (2009).
[Crossref]

K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated photons in controlled spatial modes by down conversion in nonlinear waveguides,” Opt. Lett. 26, 1367–1369 (2001).
[Crossref]

Bierlein, J. D.

J. D. Bierlein and C. B. Arweiler, “Electro-optic and dielectric properties of KTiOPO4,” Appl. Phys. Lett. 49, 917–919 (1986).
[Crossref]

Booth, M. C.

M. C. Booth, M. Atatüre, G. D. Giuseppe, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Counterpropagating entangled photons from a waveguide with periodic nonlinearity,” Phys. Rev. A 66, 023815 (2002).
[Crossref]

Bouwmeester, D.

J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Optimal generation of pulsed entangled photon pairs,” Phys. Rev. A 74, 013802 (2006).
[Crossref]

Brendel, J.

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

Chadderdon, S.

D. Perry, S. Chadderdon, R. Forber, W. C. Wang, R. Selfridge, and S. Schultz, “Multiaxis electric field sensing using slab coupled optical sensors,” App. Opt. 52, 1968–1977 (2013).
[Crossref]

Chiang, K.S.

Choi, S. K.

Y. Kang, J. Ko, S. M. Lee, S. K. Choi, B. Y. Kim, and H. S. Park, “Measurement of the entanglement between photonic spatial modes in optical fibers,” Phys. Rev. Lett. 109, 020502 (2012).
[Crossref] [PubMed]

Christ, A.

P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett. 103, 233901 (2009).
[Crossref]

Clauser, J. F.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[Crossref]

Cristofori, V.

de Micheli, M. P.

S. Tanzilli, W. Tittel, H. de Riedmatten, H. Zbinden, P. Baldi, M. P. de Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).
[Crossref]

de Riedmatten, H.

S. Tanzilli, W. Tittel, H. de Riedmatten, H. Zbinden, P. Baldi, M. P. de Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).
[Crossref]

Eberhard, P. H.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

Eckstein, A.

P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett. 103, 233901 (2009).
[Crossref]

A. Eckstein and C. Silberhorn, “Broadband frequency mode entanglement in waveguided parametric down conversion,” Opt. Lett. 33, 1825–1827 (2008).
[Crossref] [PubMed]

Fiorentino, M.

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultra-bright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69, 041801 (2004).
[Crossref]

Forber, R.

D. Perry, S. Chadderdon, R. Forber, W. C. Wang, R. Selfridge, and S. Schultz, “Multiaxis electric field sensing using slab coupled optical sensors,” App. Opt. 52, 1968–1977 (2013).
[Crossref]

Furusawa, A.

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

Ghatak, A. K.

A. K. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge University, 1989).
[Crossref]

Gisin, N.

S. Tanzilli, W. Tittel, H. de Riedmatten, H. Zbinden, P. Baldi, M. P. de Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).
[Crossref]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

Giuseppe, G. D.

M. C. Booth, M. Atatüre, G. D. Giuseppe, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Counterpropagating entangled photons from a waveguide with periodic nonlinearity,” Phys. Rev. A 66, 023815 (2002).
[Crossref]

Hermann, H.

Hodelin, J. F.

J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Optimal generation of pulsed entangled photon pairs,” Phys. Rev. A 74, 013802 (2006).
[Crossref]

Holt, R. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[Crossref]

Honjo, T.

Horne, M. A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[Crossref]

Inoue, K.

Jachura, M.

Javurek, D.

D. Javůrek, J. Svozilík, and J. Peřina, “Emission of orbital-angular-momentum-entangled photon pairs in a nonlinear ring fiber utilizing spontaneous parametric down-conversion,” Phys. Rev. A 90, 043844 (2014).
[Crossref]

Jennewein, T.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[Crossref]

Jin, W.

Kang, Y.

Y. Kang, J. Ko, S. M. Lee, S. K. Choi, B. Y. Kim, and H. S. Park, “Measurement of the entanglement between photonic spatial modes in optical fibers,” Phys. Rev. Lett. 109, 020502 (2012).
[Crossref] [PubMed]

Karpinski, M.

M. Jachura, M. Karpiński, C. Radzewicz, and K. Banaszek, “High-visibility nonclassical interference of photon pairs generated in a multimode nonlinear waveguide,” Opt. Express 22, 8624–8632 (2014).
[Crossref] [PubMed]

M. Karpiński, C. Radzewicz, and K. Banaszek, “Dispersion-based control of modal characteristics for parametric down-conversion in a multimode waveguide,” Opt. Lett. 37, 878–880 (2012).
[Crossref]

M. Karpiński, C. Radzewicz, and K. Banaszek, “Generation of spatially pure photon pairs in a multimode nonlinear waveguide using intermodal dispersion,” Proc. SPIE 8518, 85180J (2012).
[Crossref]

M. Karpiński, C. Radzewicz, and K. Banaszek, “Experimental characterization of three-wave mixing in a multi-mode nonlinear KTiOPO4 waveguide,” Appl. Phys. Lett. 94, 181105 (2009).
[Crossref]

Kato, K.

K. Kato and E. Takaoka, “Sellmeier and thermo-optic dispersion formulas for KTP,” App. Opt. 41, 5040–5044 (2002).
[Crossref]

Kawahara, K.

K. Sanaka, K. Kawahara, and T. Kuga, “New high-efficiency source of photon pairs for engineering quantum entanglement,” Phys. Rev. Lett. 86, 5620–5623 (2001).
[Crossref] [PubMed]

Khoury, G.

J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Optimal generation of pulsed entangled photon pairs,” Phys. Rev. A 74, 013802 (2006).
[Crossref]

Kim, B. Y.

Y. Kang, J. Ko, S. M. Lee, S. K. Choi, B. Y. Kim, and H. S. Park, “Measurement of the entanglement between photonic spatial modes in optical fibers,” Phys. Rev. Lett. 109, 020502 (2012).
[Crossref] [PubMed]

Ko, J.

Y. Kang, J. Ko, S. M. Lee, S. K. Choi, B. Y. Kim, and H. S. Park, “Measurement of the entanglement between photonic spatial modes in optical fibers,” Phys. Rev. Lett. 109, 020502 (2012).
[Crossref] [PubMed]

Kuga, T.

K. Sanaka, K. Kawahara, and T. Kuga, “New high-efficiency source of photon pairs for engineering quantum entanglement,” Phys. Rev. Lett. 86, 5620–5623 (2001).
[Crossref] [PubMed]

Kuklewicz, C. E.

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultra-bright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69, 041801 (2004).
[Crossref]

Kwiat, P. G.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 74, 4337–4341 (1995).
[Crossref]

Lee, S. M.

Y. Kang, J. Ko, S. M. Lee, S. K. Choi, B. Y. Kim, and H. S. Park, “Measurement of the entanglement between photonic spatial modes in optical fibers,” Phys. Rev. Lett. 109, 020502 (2012).
[Crossref] [PubMed]

Mandel, L.

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[Crossref] [PubMed]

Marcuse, D.

D. Marcuse, “Electro optic coupling between TE and TM Modes in Anisotropic Slabs,” IEEE J. Quantum Electron. QE-9, 759–767 (1975).
[Crossref]

Martin, A.

Mattle, K.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 74, 4337–4341 (1995).
[Crossref]

Messin, G.

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultra-bright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69, 041801 (2004).
[Crossref]

Mosley, P. J.

P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett. 103, 233901 (2009).
[Crossref]

O’Brien, J. L.

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

Ostrowsky, D. B.

A. Martin, V. Cristofori, P. Aboussouam, H. Hermann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “Integrated optical source of polarization entangled photons at 1310 nm,” Opt. Express 17, 1033–1041 (2009).
[Crossref] [PubMed]

S. Tanzilli, W. Tittel, H. de Riedmatten, H. Zbinden, P. Baldi, M. P. de Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).
[Crossref]

Ou, Z. Y.

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[Crossref] [PubMed]

Park, H. S.

Y. Kang, J. Ko, S. M. Lee, S. K. Choi, B. Y. Kim, and H. S. Park, “Measurement of the entanglement between photonic spatial modes in optical fibers,” Phys. Rev. Lett. 109, 020502 (2012).
[Crossref] [PubMed]

Perina, J.

D. Javůrek, J. Svozilík, and J. Peřina, “Emission of orbital-angular-momentum-entangled photon pairs in a nonlinear ring fiber utilizing spontaneous parametric down-conversion,” Phys. Rev. A 90, 043844 (2014).
[Crossref]

Perry, D.

D. Perry, S. Chadderdon, R. Forber, W. C. Wang, R. Selfridge, and S. Schultz, “Multiaxis electric field sensing using slab coupled optical sensors,” App. Opt. 52, 1968–1977 (2013).
[Crossref]

Radzewicz, C.

M. Jachura, M. Karpiński, C. Radzewicz, and K. Banaszek, “High-visibility nonclassical interference of photon pairs generated in a multimode nonlinear waveguide,” Opt. Express 22, 8624–8632 (2014).
[Crossref] [PubMed]

M. Karpiński, C. Radzewicz, and K. Banaszek, “Dispersion-based control of modal characteristics for parametric down-conversion in a multimode waveguide,” Opt. Lett. 37, 878–880 (2012).
[Crossref]

M. Karpiński, C. Radzewicz, and K. Banaszek, “Generation of spatially pure photon pairs in a multimode nonlinear waveguide using intermodal dispersion,” Proc. SPIE 8518, 85180J (2012).
[Crossref]

M. Karpiński, C. Radzewicz, and K. Banaszek, “Experimental characterization of three-wave mixing in a multi-mode nonlinear KTiOPO4 waveguide,” Appl. Phys. Lett. 94, 181105 (2009).
[Crossref]

Saleh, B. E. A.

M. F. Saleh, B. E. A. Saleh, and M. C. Teich, “Modal, spectral, and polarization entanglement in guided-wave parametric down-conversion,” Phys. Rev. A 79, 053842 (2009).
[Crossref]

M. C. Booth, M. Atatüre, G. D. Giuseppe, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Counterpropagating entangled photons from a waveguide with periodic nonlinearity,” Phys. Rev. A 66, 023815 (2002).
[Crossref]

Saleh, M. F.

M. F. Saleh, B. E. A. Saleh, and M. C. Teich, “Modal, spectral, and polarization entanglement in guided-wave parametric down-conversion,” Phys. Rev. A 79, 053842 (2009).
[Crossref]

Sanaka, K.

K. Sanaka, K. Kawahara, and T. Kuga, “New high-efficiency source of photon pairs for engineering quantum entanglement,” Phys. Rev. Lett. 86, 5620–5623 (2001).
[Crossref] [PubMed]

Schultz, S.

D. Perry, S. Chadderdon, R. Forber, W. C. Wang, R. Selfridge, and S. Schultz, “Multiaxis electric field sensing using slab coupled optical sensors,” App. Opt. 52, 1968–1977 (2013).
[Crossref]

Selfridge, R.

D. Perry, S. Chadderdon, R. Forber, W. C. Wang, R. Selfridge, and S. Schultz, “Multiaxis electric field sensing using slab coupled optical sensors,” App. Opt. 52, 1968–1977 (2013).
[Crossref]

Sergienko, A. V.

M. C. Booth, M. Atatüre, G. D. Giuseppe, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Counterpropagating entangled photons from a waveguide with periodic nonlinearity,” Phys. Rev. A 66, 023815 (2002).
[Crossref]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 74, 4337–4341 (1995).
[Crossref]

Shapiro, J. H.

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultra-bright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69, 041801 (2004).
[Crossref]

Shih, Y.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 74, 4337–4341 (1995).
[Crossref]

Shimony, A.

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[Crossref]

Silberhorn, C.

P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett. 103, 233901 (2009).
[Crossref]

A. Eckstein and C. Silberhorn, “Broadband frequency mode entanglement in waveguided parametric down conversion,” Opt. Lett. 33, 1825–1827 (2008).
[Crossref] [PubMed]

Simon, C.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[Crossref]

Sohler, W.

Svozilík, J.

D. Javůrek, J. Svozilík, and J. Peřina, “Emission of orbital-angular-momentum-entangled photon pairs in a nonlinear ring fiber utilizing spontaneous parametric down-conversion,” Phys. Rev. A 90, 043844 (2014).
[Crossref]

Takaoka, E.

K. Kato and E. Takaoka, “Sellmeier and thermo-optic dispersion formulas for KTP,” App. Opt. 41, 5040–5044 (2002).
[Crossref]

Takesue, H.

Tanzilli, S.

A. Martin, V. Cristofori, P. Aboussouam, H. Hermann, W. Sohler, D. B. Ostrowsky, O. Alibart, and S. Tanzilli, “Integrated optical source of polarization entangled photons at 1310 nm,” Opt. Express 17, 1033–1041 (2009).
[Crossref] [PubMed]

S. Tanzilli, W. Tittel, H. de Riedmatten, H. Zbinden, P. Baldi, M. P. de Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).
[Crossref]

Teich, M. C.

M. F. Saleh, B. E. A. Saleh, and M. C. Teich, “Modal, spectral, and polarization entanglement in guided-wave parametric down-conversion,” Phys. Rev. A 79, 053842 (2009).
[Crossref]

M. C. Booth, M. Atatüre, G. D. Giuseppe, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Counterpropagating entangled photons from a waveguide with periodic nonlinearity,” Phys. Rev. A 66, 023815 (2002).
[Crossref]

Thyagarajan, K.

A. K. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge University, 1989).
[Crossref]

Tittel, W.

S. Tanzilli, W. Tittel, H. de Riedmatten, H. Zbinden, P. Baldi, M. P. de Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).
[Crossref]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

Torres, J. P.

J. P. Torres, K. Banaszek, and I. A. Walmsley, “Engineering nonlinear optic sources of photonic entanglement,” Prog. Opt. 56, 227–331 (2011).
[Crossref]

U’Ren, A. B.

Vuckovic, Jelena

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

Waks, E.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

Walmsley, I. A.

J. P. Torres, K. Banaszek, and I. A. Walmsley, “Engineering nonlinear optic sources of photonic entanglement,” Prog. Opt. 56, 227–331 (2011).
[Crossref]

K. Banaszek, A. B. U’Ren, and I. A. Walmsley, “Generation of correlated photons in controlled spatial modes by down conversion in nonlinear waveguides,” Opt. Lett. 26, 1367–1369 (2001).
[Crossref]

Wang, W. C.

D. Perry, S. Chadderdon, R. Forber, W. C. Wang, R. Selfridge, and S. Schultz, “Multiaxis electric field sensing using slab coupled optical sensors,” App. Opt. 52, 1968–1977 (2013).
[Crossref]

Weihs, G.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[Crossref]

Weinfurter, H.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[Crossref]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 74, 4337–4341 (1995).
[Crossref]

White, A. G.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

Wong, F. N. C.

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultra-bright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69, 041801 (2004).
[Crossref]

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

A. Yariv and P. Yeh, Photonics (Oxford University, 2007).

Yeh, P.

A. Yariv and P. Yeh, Photonics (Oxford University, 2007).

Zbinden, H.

S. Tanzilli, W. Tittel, H. de Riedmatten, H. Zbinden, P. Baldi, M. P. de Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).
[Crossref]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

Zeilinger, A.

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[Crossref]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 74, 4337–4341 (1995).
[Crossref]

App. Opt. (2)

D. Perry, S. Chadderdon, R. Forber, W. C. Wang, R. Selfridge, and S. Schultz, “Multiaxis electric field sensing using slab coupled optical sensors,” App. Opt. 52, 1968–1977 (2013).
[Crossref]

K. Kato and E. Takaoka, “Sellmeier and thermo-optic dispersion formulas for KTP,” App. Opt. 41, 5040–5044 (2002).
[Crossref]

Appl. Phys. Lett. (2)

J. D. Bierlein and C. B. Arweiler, “Electro-optic and dielectric properties of KTiOPO4,” Appl. Phys. Lett. 49, 917–919 (1986).
[Crossref]

M. Karpiński, C. Radzewicz, and K. Banaszek, “Experimental characterization of three-wave mixing in a multi-mode nonlinear KTiOPO4 waveguide,” Appl. Phys. Lett. 94, 181105 (2009).
[Crossref]

Eur. Phys. J. D (1)

S. Tanzilli, W. Tittel, H. de Riedmatten, H. Zbinden, P. Baldi, M. P. de Micheli, D. B. Ostrowsky, and N. Gisin, “PPLN waveguide for quantum communication,” Eur. Phys. J. D 18, 155–160 (2002).
[Crossref]

IEEE J. Quantum Electron. (2)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[Crossref]

D. Marcuse, “Electro optic coupling between TE and TM Modes in Anisotropic Slabs,” IEEE J. Quantum Electron. QE-9, 759–767 (1975).
[Crossref]

Nature Photon. (1)

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

Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. A (6)

M. F. Saleh, B. E. A. Saleh, and M. C. Teich, “Modal, spectral, and polarization entanglement in guided-wave parametric down-conversion,” Phys. Rev. A 79, 053842 (2009).
[Crossref]

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A 60, R773–R776 (1999).
[Crossref]

M. Fiorentino, G. Messin, C. E. Kuklewicz, F. N. C. Wong, and J. H. Shapiro, “Generation of ultra-bright tunable polarization entanglement without spatial, spectral, or temporal constraints,” Phys. Rev. A 69, 041801 (2004).
[Crossref]

J. F. Hodelin, G. Khoury, and D. Bouwmeester, “Optimal generation of pulsed entangled photon pairs,” Phys. Rev. A 74, 013802 (2006).
[Crossref]

M. C. Booth, M. Atatüre, G. D. Giuseppe, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Counterpropagating entangled photons from a waveguide with periodic nonlinearity,” Phys. Rev. A 66, 023815 (2002).
[Crossref]

D. Javůrek, J. Svozilík, and J. Peřina, “Emission of orbital-angular-momentum-entangled photon pairs in a nonlinear ring fiber utilizing spontaneous parametric down-conversion,” Phys. Rev. A 90, 043844 (2014).
[Crossref]

Phys. Rev. Lett. (8)

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969).
[Crossref]

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[Crossref] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 74, 4337–4341 (1995).
[Crossref]

G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, and A. Zeilinger, “Violation of Bell’s inequality under strict Einstein locality conditions,” Phys. Rev. Lett. 81, 5039–5043 (1998).
[Crossref]

Y. Kang, J. Ko, S. M. Lee, S. K. Choi, B. Y. Kim, and H. S. Park, “Measurement of the entanglement between photonic spatial modes in optical fibers,” Phys. Rev. Lett. 109, 020502 (2012).
[Crossref] [PubMed]

P. J. Mosley, A. Christ, A. Eckstein, and C. Silberhorn, “Direct measurement of the spatial-spectral structure of waveguided parametric down-conversion,” Phys. Rev. Lett. 103, 233901 (2009).
[Crossref]

J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, “Pulsed energy-time entangled twin-photon source for quantum communication,” Phys. Rev. Lett. 82, 2594–2597 (1999).
[Crossref]

K. Sanaka, K. Kawahara, and T. Kuga, “New high-efficiency source of photon pairs for engineering quantum entanglement,” Phys. Rev. Lett. 86, 5620–5623 (2001).
[Crossref] [PubMed]

Proc. SPIE (1)

M. Karpiński, C. Radzewicz, and K. Banaszek, “Generation of spatially pure photon pairs in a multimode nonlinear waveguide using intermodal dispersion,” Proc. SPIE 8518, 85180J (2012).
[Crossref]

Prog. Opt. (1)

J. P. Torres, K. Banaszek, and I. A. Walmsley, “Engineering nonlinear optic sources of photonic entanglement,” Prog. Opt. 56, 227–331 (2011).
[Crossref]

Other (2)

A. K. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge University, 1989).
[Crossref]

A. Yariv and P. Yeh, Photonics (Oxford University, 2007).

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Figures (6)

Fig. 1
Fig. 1 (a) Electric field distributions for transverse spatial modes 00 and 10 for horizontal H and vertical V polarizations in an exemplary KTP waveguide used for the numerical example presented in Sec. 4 of this paper. (b) Rotation of the spatial mode basis required to test the CHSH inequality. Rotated modes |θ〉 and | θ + π 2 are illustrated with distributions depicted for θ = 30°.
Fig. 2
Fig. 2 Top view of the waveguide design for generating and testing of spatially mode entangled state. Region (a) is a multimode channel waveguide with a single QPM grating for generating spatially mode entangled states and region (b) is a two-moded channel waveguide with two asymmetric electrode patterns of different periods to independently control coupling among the H and V polarized spatial modes. Part (c) is an asymmetric Y-splitter separating modes 00 and 10 into distinct output ports.
Fig. 3
Fig. 3 (a) Cross-sectional view of the region Fig. 2(b) depicting the geometry of the device. Here, 2a is the separation between the two electrodes, b and c corresponds to the width and the depth of the two mode channel waveguide respectively and d is the separation of midpoint of two electrodes (dashed line) with respect to the fixed center of the waveguide. (b) Optimized parameters of the device ensuring an efficient and controlable coupling between the 10 and 00 modes, while preserving both two-mode operation as well as polarization selectivity.
Fig. 4
Fig. 4 Spectral distributions of the two SPDC processes used to prepare a pair of photons entangled in transverse spatial modes.
Fig. 5
Fig. 5 (a) Variation of κH with d for 1 volt for different value of a, (b) Variation of κV with d for 1 volt for different value of a.
Fig. 6
Fig. 6 Variation of normalized power for vertically polarized 00(10) and horizontally polarized 10(00) mode with voltage, corresponding to the spatial frequency (a) KH = 2π/102.5µm and (b) KV = 2π/105.1µm.

Equations (34)

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1 : 10 p 00 H + 10 V 2 : 10 p 10 H + 00 V .
| Ψ = 1 2 ( | 00 H , 10 V + | 10 H , 00 V ) .
| θ = cos θ | 00 + sin θ | 10 | θ + π 2 = sin θ | 00 + cos θ | 10 ,
( A H ( L ) B H ( L ) ) = ( cos ( κ H L ) sin ( κ H L ) sin ( κ H L ) cos ( κ H L ) ) ( A H ( 0 ) B H ( 0 ) ) ,
( A V ( L ) B V ( L ) ) = ( cos ( κ V L ) sin ( κ V L ) sin ( κ V L ) cos ( κ V L ) ) ( A V ( 0 ) B V ( 0 ) ) .
E ( θ 1 , θ 2 ) = P 00 ( θ 1 , θ 2 ) + P 11 ( θ 1 , θ 2 ) P 01 ( θ 1 , θ 2 ) P 10 ( θ 1 , θ 2 ) .
S = E ( θ 1 , θ 2 ) + E ( θ 1 , θ 2 ) + E ( θ 1 , θ 2 ) E ( θ 1 , θ 2 ) ,
x 2 ( n x + Δ n ˜ x ) 2 + y 2 ( n y + Δ n ˜ y ) 2 + z 2 ( n z + Δ n ˜ z ) 2 + 2 y z r 42 E y + 2 x z r 51 E x = 1 ,
( 0 0 r 13 0 0 r 23 0 0 r 33 0 r 42 0 r 51 0 0 0 0 0 ) .
Φ y ( z ) ( x , y , z ) = E 00 H ( V ) ( x , y , z ) + E 10 H ( V ) ( x , y , z ) = = A H ( V ) ( x ) ψ 00 H ( V ) ( y , z ) e i ( ω t β 00 H ( V ) x ) + B H ( V ) ( x ) ψ 10 H ( V ) ( y , z ) e i ( ω t β 10 H ( V ) x ) ,
2 ψ 00 H ( V ) y 2 + 2 ψ 00 H ( V ) z 2 + [ k 0 2 n y ( z ) 2 ( y , z ) ( β 00 H ( V ) ) 2 ] ψ 00 H ( V ) = 0
2 ψ 10 H ( V ) y 2 + 2 ψ 10 H ( V ) z 2 + [ k 0 2 n y ( z ) 2 ( y , z ) ( β 10 H ( V ) ) 2 ] ψ 10 H ( V ) = 0.
n ˜ y ( z ) 2 ( x , y , z ) n y ( z ) 2 ( y , z ) + 2 n y ( z ) ( y , z ) Δ n y ( z ) ( x , y , z ) .
Δ n y ( z ) ( x , y , z ) = Δ n ˜ y ( z ) ( y , z ) sin ( K H ( V ) x ) ,
A H ( V ) x = κ 12 H ( V ) B H ( V ) e i ( Δ β H ( V ) K H ( V ) ) x
B H ( V ) x = κ 21 H ( V ) A H ( V ) e i ( Δ β H ( V ) K H ( V ) ) x ,
κ 12 H ( V ) = k 0 2 2 β 00 H ( V ) ( ψ 00 H ( V ) ) * n y ( z ) ( y , z ) Δ n ˜ y ( z ) ( y , z ) ( ψ 10 H ( V ) ) d y d z ,
κ 21 H ( V ) = k 0 2 2 β 10 H ( V ) ( ψ 10 H ( V ) ) * n y ( z ) ( y , z ) Δ n ˜ y ( z ) ( y , z ) ( ψ 00 H ( V ) ) d y d z ,
Δ β H ( V ) = β 00 H ( V ) β 10 H ( V ) .
( A H ( V ) ( x ) B H ( V ) ( x ) ) = ( [ cos ( γ H ( V ) x ) i δ H ( V ) γ H ( V ) sin ( γ H ( V ) x ) ] e i δ H ( V ) x , [ κ H ( V ) γ H ( V ) sin ( γ H ( V ) x ) ] e i δ H ( V ) x [ κ H ( V ) γ H ( V ) sin ( γ H ( V ) x ) ] e i δ H ( V ) x , [ cos ( γ H ( V ) x ) + i δ H ( V ) γ H ( V ) sin ( γ H ( V ) x ) ] e i δ H ( V ) x ) ( A H ( V ) ( 0 ) B H ( V ) ( 0 ) ) ,
δ H ( V ) = Δ β H ( V ) K H ( V ) 2 , κ H ( V ) = κ 12 H ( V ) κ 21 H ( V ) , ( γ H ( V ) ) 2 = ( κ H ( V ) ) 2 + ( δ H ( V ) ) 2 .
A H ( x = 0 ) = 1 & B H ( x = 0 ) = 0 ; B V ( x = 0 ) = 1 & A V ( x = 0 ) = 0.
B H ( x = 0 ) = 1 & A H ( x = 0 ) = 0 ; A V ( x = 0 ) = 1 & B V ( x = 0 ) = 0.
A H ( x ) = [ cos ( γ H x ) i δ H γ H sin ( γ H x ) ] e i δ H x ; B H ( x ) = [ κ H γ H sin ( γ H x ) ] e i δ H x
A V ( x ) = [ κ V γ V sin ( γ V x ) ] e i δ V x ; B V ( x ) = [ cos ( γ V x ) + i δ V γ V sin ( γ V x ) ] e i δ v x .
P H 00 ( L ) = | A H | 2 = cos 2 ( γ H L ) + ( δ H γ H ) 2 sin 2 ( γ H L ) ; P H 10 ( L ) = | B H | 2 = ( κ H γ H ) 2 sin 2 ( γ H L )
P V 00 ( L ) = | A V | 2 = ( κ V γ V ) 2 sin 2 ( γ V L ) ; P V 10 ( L ) = | B V | 2 = cos 2 ( γ V L ) + ( δ V γ V ) 2 sin 2 ( γ V L ) .
P H 00 ( L ) = | A H | 2 ( κ H γ H ) 2 sin 2 ( γ H L ) ; P H 10 ( L ) = | B H | 2 = cos 2 ( γ H L ) + ( δ H γ H ) 2 sin 2 ( γ H L ) P V 00 ( L ) = | A V | 2 = cos 2 ( γ V L ) + ( δ V γ V ) 2 sin 2 ( γ V L ) ; P V 10 ( L ) = | B V | 2 = ( κ V γ V ) 2 sin 2 ( γ V L ) .
( A H ( V ) ( L ) B H ( V ) ( L ) ) = ( cos ( κ H ( V ) L ) sin ( κ H ( V ) L ) sin ( κ H ( V ) L ) cos ( κ H ( V ) L ) ) ( A H ( V ) ( 0 ) B H ( V ) ( 0 ) ) .
| θ H ( V ) = cos θ | 00 H ( V ) + sin θ | 10 H ( V )
| ( θ + π 2 ) H ( V ) = sin θ | 00 H ( V ) + cos θ | 10 H ( V ) .
P m n ( θ 1 , θ 2 ) = ( θ 1 + m π 2 ) H , ( θ 2 + n π 2 ) V | ρ ^ | ( θ 1 + m π 2 ) H , ( θ 2 + n π 2 ) V ,
ρ ^ = w | 00 H , 10 V 00 H , 10 V | + ( 1 w ) | 10 H , 00 V 10 H , 00 V | + v ( | 00 H , 10 V 10 H , 00 V | + | 10 H , 00 V 00 H , 10 V | )
E z ( y , z ) = V π 2 [ ( a 2 + z 2 y 2 ) + { a 4 + y 4 + z 4 + 2 y 2 z 2 + 2 a 2 ( z 2 y 2 ) } 1 / 2 ] 1 / 2 [ a 4 + y 4 + z 4 + 2 y 2 z 2 + 2 a 2 ( z 2 y 2 ) ] 1 / 2 ,

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