Abstract

Coherence properties of the photon pair generated via spontaneous parametric down-conversion pumped by a multi-mode cw diode laser are studied with a Mach-Zehnder interferometer. Each photon of the pair enters a different input port of the interferometer and the biphoton coherence properties are studied with a two-photon detector placed at one output port. When the photon pair simultaneously enters the interferometer, periodic recurrence of the biphoton de Broglie wave packet is observed, closely resembling the coherence properties of the pump diode laser. With non-zero delays between the photons at the input ports, biphoton interference exhibits the same periodic recurrence but the wave packet shapes are shown to be dependent on both the input delay as well as the interferometer delay. These properties could be useful for building engineered entangled photon sources based on diode laser-pumped spontaneous parametric down-conversion.

© 2009 Optical Society of America

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  1. J. P. Dowling and G. J. Milburn, "Quantum technology: the second quantum revolution," Phil. Trans. R. Soc. Lond. A 361, 1655-1674 (2003).
    [CrossRef]
  2. N. Gisin, G. Ribordy,W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
    [CrossRef]
  3. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-174 (2007).
    [CrossRef]
  4. Y. H. Shih and C. O. Alley, "New Type of Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quants Produced by Optical Parametric Down Conversion," Phys. Rev. Lett. 61, 2921-2924 (1988).
    [CrossRef] [PubMed]
  5. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New High-Intensity Source of Polarization-Entangled Photon Pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
    [CrossRef] [PubMed]
  6. Y.-H. Kim, S. P. Kulik, M. V. Chekhova, W. P. Grice, and Y. Shih, "Experimental entanglement concentration and universal Bell-state synthesizer," Phys. Rev. A 67, 010301(R) (2003).
    [CrossRef]
  7. J. G. Rarity and P. R. Tapster, "Experimental Violation of Bell’s Inequality Based on Phase and Momentum," Phys. Rev. Lett. 64, 2495-2498 (1990).
    [CrossRef] [PubMed]
  8. D. V. Strekalov, T. B. Pittman, A. V. Sergienko, Y. H. Shih, and P. G. Kwiat, "Postselection-free energy-time entanglement," Phys. Rev. A 54, R1-R4 (1996).
    [CrossRef] [PubMed]
  9. W. P. Grice, A. B. U’Ren, and I. A. Walmsley, "Elimanating frequency and space-tie corelations in multiphoton states," Phys. Rev. A 64, 063815 (2001).
    [CrossRef]
  10. Y.-H. Kim and W. P. Grice, "Generation of pulsed polarization-entangled two-photon state via temporal and spectral engineering," J. Mod. Opt. 49, 2309-2323 (2002).
    [CrossRef]
  11. A. Valencia, A. Cere, X. Shi, G. Molina-Terriza, and J. P. Torres, "Shaping theWaveform of Entangled Photons," Phys. Rev. Lett. 99, 243601 (2007).
    [CrossRef]
  12. P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A.Walmsley, "Heralded Generation of Ultrafast Single Photons in Pure Quantum States," Phys. Rev. Lett. 100, 133601 (2008).
    [CrossRef] [PubMed]
  13. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773-R776 (1999).
    [CrossRef]
  14. Y.-H. Kim, M. V. Chekhova, S. P. Kulik, M. H. Rubin, and Y. Shih, "Interferometric Bell-state preparation using femtosecond-pulse-pumped spontaneous parametric down-conversion," Phys. Rev. A 63, 062301 (2001).
    [CrossRef]
  15. Y.-H. Kim and W. P. Grice, "Measurement of the spectral properties of the two-photon state generated via type II spontaneous parametric downconversion," Opt. Lett. 30, 908-910 (2005).
    [CrossRef] [PubMed]
  16. A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P Kulik, "Biphoton interference with a multimode pump," Phys. Rev. A 63, 053801 (2001).
    [CrossRef]
  17. The Wiener-Khinchine theorem states that the spectral power density of an optical field is related to its autocorrelation. They are, in fact, a Fourier transform pair. It is not difficult to show that the following relation holds, ⊗⌊FWHM = (4ln2/) 2 center/LFWHM. Here ⊗FWHM is the FWHM bandwidth of the field, center is the central wavelength of the laser (405 nm), and LFWHM is the FWHM width of the autocorrelation (interferogram) in Fig. 2.
  18. J. W. Goodman, Statistical optics (Wiley, New York, 1985), p. 230.
  19. S.-Y. Baek, O. Kwon, and Y.-H. Kim, "High-Resolution Mode-Spacing Measurement of the Blue-Violet Diode Laser Using Interference of Felds Created with Time Delays Greater than the Coherence Time," Jpn. J. Appl. Phys. 46, 7720-7723 (2007).
    [CrossRef]
  20. Y. J. Lu, R. L. Campbell, and Z. Y. Ou, "Mode-locked two-photon states," Phys. Rev. Lett. 91, 163602 (2003).
    [CrossRef] [PubMed]
  21. M. A. Sagioro, C. Olindo, C. H. Monken, and S. Pádua, "Time control of two-photon interference," Phys. Rev. A 69, 053817 (2004).
    [CrossRef]
  22. A. Zavatta, S. Viciani, and M. Bellini, "Recurrent fourth-order interference dips and peaks with a comblike two-photon entangled state," Phys. Rev. A 70, 023806 (2004).
    [CrossRef]
  23. C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
    [CrossRef] [PubMed]
  24. J. Jacobson, G. Bjork, I. Chuang, and Y. Yamamoto, "Photonic de Broglie waves," Phys. Rev. Lett. 74, 4835-4838 (1995).
    [CrossRef] [PubMed]
  25. K. Edamatsu, R. Shimizu, and T. Itoh, "Measurement of the Photonic de BroglieWavelength of Entangled Photon Pairs Generated by Spontaneous Parametric Down-Conversion," Phys. Rev. Lett. 89, 213601 (2002).
    [CrossRef] [PubMed]
  26. J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "Two-photon interference in a Mach-Zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
    [CrossRef] [PubMed]
  27. Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical fourth-order interference," Phys. Rev. A 42, 2957-2965 (1990).
    [CrossRef] [PubMed]
  28. 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]
  29. Y.-H. Kim, V. Berardi, M. V. Chekhova, A. Garuccio, and Y. H. Shih, "Temporal indistinguishability and quantum interference," Phys. Rev. A 62, 43820 (2000).
    [CrossRef]
  30. S.-Y. Baek and Y.-H. Kim, "Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion," Phys. Rev. A 77, 043807 (2008).
    [CrossRef]
  31. We have measured the transmission function of the interference filters used in this experiment with an Agilent 8453 UV/VIS spectro-photometer. We have found that the transmission function is indeed very close Gaussian centered at 810 nm as assumed in eq. (10).

2008 (2)

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A.Walmsley, "Heralded Generation of Ultrafast Single Photons in Pure Quantum States," Phys. Rev. Lett. 100, 133601 (2008).
[CrossRef] [PubMed]

S.-Y. Baek and Y.-H. Kim, "Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion," Phys. Rev. A 77, 043807 (2008).
[CrossRef]

2007 (3)

A. Valencia, A. Cere, X. Shi, G. Molina-Terriza, and J. P. Torres, "Shaping theWaveform of Entangled Photons," Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

S.-Y. Baek, O. Kwon, and Y.-H. Kim, "High-Resolution Mode-Spacing Measurement of the Blue-Violet Diode Laser Using Interference of Felds Created with Time Delays Greater than the Coherence Time," Jpn. J. Appl. Phys. 46, 7720-7723 (2007).
[CrossRef]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

2005 (1)

2004 (2)

M. A. Sagioro, C. Olindo, C. H. Monken, and S. Pádua, "Time control of two-photon interference," Phys. Rev. A 69, 053817 (2004).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, "Recurrent fourth-order interference dips and peaks with a comblike two-photon entangled state," Phys. Rev. A 70, 023806 (2004).
[CrossRef]

2003 (2)

J. P. Dowling and G. J. Milburn, "Quantum technology: the second quantum revolution," Phil. Trans. R. Soc. Lond. A 361, 1655-1674 (2003).
[CrossRef]

Y. J. Lu, R. L. Campbell, and Z. Y. Ou, "Mode-locked two-photon states," Phys. Rev. Lett. 91, 163602 (2003).
[CrossRef] [PubMed]

2002 (3)

Y.-H. Kim and W. P. Grice, "Generation of pulsed polarization-entangled two-photon state via temporal and spectral engineering," J. Mod. Opt. 49, 2309-2323 (2002).
[CrossRef]

N. Gisin, G. Ribordy,W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

K. Edamatsu, R. Shimizu, and T. Itoh, "Measurement of the Photonic de BroglieWavelength of Entangled Photon Pairs Generated by Spontaneous Parametric Down-Conversion," Phys. Rev. Lett. 89, 213601 (2002).
[CrossRef] [PubMed]

2001 (3)

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, "Elimanating frequency and space-tie corelations in multiphoton states," Phys. Rev. A 64, 063815 (2001).
[CrossRef]

Y.-H. Kim, M. V. Chekhova, S. P. Kulik, M. H. Rubin, and Y. Shih, "Interferometric Bell-state preparation using femtosecond-pulse-pumped spontaneous parametric down-conversion," Phys. Rev. A 63, 062301 (2001).
[CrossRef]

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P Kulik, "Biphoton interference with a multimode pump," Phys. Rev. A 63, 053801 (2001).
[CrossRef]

2000 (1)

Y.-H. Kim, V. Berardi, M. V. Chekhova, A. Garuccio, and Y. H. Shih, "Temporal indistinguishability and quantum interference," Phys. Rev. A 62, 43820 (2000).
[CrossRef]

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 polarizationentangled photons," Phys. Rev. A 60, R773-R776 (1999).
[CrossRef]

1996 (1)

D. V. Strekalov, T. B. Pittman, A. V. Sergienko, Y. H. Shih, and P. G. Kwiat, "Postselection-free energy-time entanglement," Phys. Rev. A 54, R1-R4 (1996).
[CrossRef] [PubMed]

1995 (2)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New High-Intensity Source of Polarization-Entangled Photon Pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

J. Jacobson, G. Bjork, I. Chuang, and Y. Yamamoto, "Photonic de Broglie waves," Phys. Rev. Lett. 74, 4835-4838 (1995).
[CrossRef] [PubMed]

1990 (3)

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "Two-photon interference in a Mach-Zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical fourth-order interference," Phys. Rev. A 42, 2957-2965 (1990).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, "Experimental Violation of Bell’s Inequality Based on Phase and Momentum," Phys. Rev. Lett. 64, 2495-2498 (1990).
[CrossRef] [PubMed]

1988 (1)

Y. H. Shih and C. O. Alley, "New Type of Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quants Produced by Optical Parametric Down Conversion," Phys. Rev. Lett. 61, 2921-2924 (1988).
[CrossRef] [PubMed]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef] [PubMed]

Alley, C. O.

Y. H. Shih and C. O. Alley, "New Type of Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quants Produced by Optical Parametric Down Conversion," Phys. Rev. Lett. 61, 2921-2924 (1988).
[CrossRef] [PubMed]

Appelbaum, I.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773-R776 (1999).
[CrossRef]

Baek, S.-Y.

S.-Y. Baek and Y.-H. Kim, "Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion," Phys. Rev. A 77, 043807 (2008).
[CrossRef]

S.-Y. Baek, O. Kwon, and Y.-H. Kim, "High-Resolution Mode-Spacing Measurement of the Blue-Violet Diode Laser Using Interference of Felds Created with Time Delays Greater than the Coherence Time," Jpn. J. Appl. Phys. 46, 7720-7723 (2007).
[CrossRef]

Bellini, M.

A. Zavatta, S. Viciani, and M. Bellini, "Recurrent fourth-order interference dips and peaks with a comblike two-photon entangled state," Phys. Rev. A 70, 023806 (2004).
[CrossRef]

Berardi, V.

Y.-H. Kim, V. Berardi, M. V. Chekhova, A. Garuccio, and Y. H. Shih, "Temporal indistinguishability and quantum interference," Phys. Rev. A 62, 43820 (2000).
[CrossRef]

Bjork, G.

J. Jacobson, G. Bjork, I. Chuang, and Y. Yamamoto, "Photonic de Broglie waves," Phys. Rev. Lett. 74, 4835-4838 (1995).
[CrossRef] [PubMed]

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]

Burlakov, A. V.

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P Kulik, "Biphoton interference with a multimode pump," Phys. Rev. A 63, 053801 (2001).
[CrossRef]

Campbell, R. L.

Y. J. Lu, R. L. Campbell, and Z. Y. Ou, "Mode-locked two-photon states," Phys. Rev. Lett. 91, 163602 (2003).
[CrossRef] [PubMed]

Campos, R. A.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "Two-photon interference in a Mach-Zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Cere, A.

A. Valencia, A. Cere, X. Shi, G. Molina-Terriza, and J. P. Torres, "Shaping theWaveform of Entangled Photons," Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

Chekhova, M. V.

Y.-H. Kim, M. V. Chekhova, S. P. Kulik, M. H. Rubin, and Y. Shih, "Interferometric Bell-state preparation using femtosecond-pulse-pumped spontaneous parametric down-conversion," Phys. Rev. A 63, 062301 (2001).
[CrossRef]

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P Kulik, "Biphoton interference with a multimode pump," Phys. Rev. A 63, 053801 (2001).
[CrossRef]

Y.-H. Kim, V. Berardi, M. V. Chekhova, A. Garuccio, and Y. H. Shih, "Temporal indistinguishability and quantum interference," Phys. Rev. A 62, 43820 (2000).
[CrossRef]

Chuang, I.

J. Jacobson, G. Bjork, I. Chuang, and Y. Yamamoto, "Photonic de Broglie waves," Phys. Rev. Lett. 74, 4835-4838 (1995).
[CrossRef] [PubMed]

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

J. P. Dowling and G. J. Milburn, "Quantum technology: the second quantum revolution," Phil. Trans. R. Soc. Lond. A 361, 1655-1674 (2003).
[CrossRef]

Eberhard, P. H.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773-R776 (1999).
[CrossRef]

Edamatsu, K.

K. Edamatsu, R. Shimizu, and T. Itoh, "Measurement of the Photonic de BroglieWavelength of Entangled Photon Pairs Generated by Spontaneous Parametric Down-Conversion," Phys. Rev. Lett. 89, 213601 (2002).
[CrossRef] [PubMed]

Garuccio, A.

Y.-H. Kim, V. Berardi, M. V. Chekhova, A. Garuccio, and Y. H. Shih, "Temporal indistinguishability and quantum interference," Phys. Rev. A 62, 43820 (2000).
[CrossRef]

Gisin, N.

N. Gisin, G. Ribordy,W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (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]

Grice, W. P.

Y.-H. Kim and W. P. Grice, "Measurement of the spectral properties of the two-photon state generated via type II spontaneous parametric downconversion," Opt. Lett. 30, 908-910 (2005).
[CrossRef] [PubMed]

Y.-H. Kim and W. P. Grice, "Generation of pulsed polarization-entangled two-photon state via temporal and spectral engineering," J. Mod. Opt. 49, 2309-2323 (2002).
[CrossRef]

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, "Elimanating frequency and space-tie corelations in multiphoton states," Phys. Rev. A 64, 063815 (2001).
[CrossRef]

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef] [PubMed]

Itoh, T.

K. Edamatsu, R. Shimizu, and T. Itoh, "Measurement of the Photonic de BroglieWavelength of Entangled Photon Pairs Generated by Spontaneous Parametric Down-Conversion," Phys. Rev. Lett. 89, 213601 (2002).
[CrossRef] [PubMed]

Jacobson, J.

J. Jacobson, G. Bjork, I. Chuang, and Y. Yamamoto, "Photonic de Broglie waves," Phys. Rev. Lett. 74, 4835-4838 (1995).
[CrossRef] [PubMed]

Jakeman, E.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "Two-photon interference in a Mach-Zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Karabutova, O. A.

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P Kulik, "Biphoton interference with a multimode pump," Phys. Rev. A 63, 053801 (2001).
[CrossRef]

Kim, Y.-H.

S.-Y. Baek and Y.-H. Kim, "Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion," Phys. Rev. A 77, 043807 (2008).
[CrossRef]

S.-Y. Baek, O. Kwon, and Y.-H. Kim, "High-Resolution Mode-Spacing Measurement of the Blue-Violet Diode Laser Using Interference of Felds Created with Time Delays Greater than the Coherence Time," Jpn. J. Appl. Phys. 46, 7720-7723 (2007).
[CrossRef]

Y.-H. Kim and W. P. Grice, "Measurement of the spectral properties of the two-photon state generated via type II spontaneous parametric downconversion," Opt. Lett. 30, 908-910 (2005).
[CrossRef] [PubMed]

Y.-H. Kim and W. P. Grice, "Generation of pulsed polarization-entangled two-photon state via temporal and spectral engineering," J. Mod. Opt. 49, 2309-2323 (2002).
[CrossRef]

Y.-H. Kim, M. V. Chekhova, S. P. Kulik, M. H. Rubin, and Y. Shih, "Interferometric Bell-state preparation using femtosecond-pulse-pumped spontaneous parametric down-conversion," Phys. Rev. A 63, 062301 (2001).
[CrossRef]

Y.-H. Kim, V. Berardi, M. V. Chekhova, A. Garuccio, and Y. H. Shih, "Temporal indistinguishability and quantum interference," Phys. Rev. A 62, 43820 (2000).
[CrossRef]

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

Kulik, S. P

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P Kulik, "Biphoton interference with a multimode pump," Phys. Rev. A 63, 053801 (2001).
[CrossRef]

Kulik, S. P.

Y.-H. Kim, M. V. Chekhova, S. P. Kulik, M. H. Rubin, and Y. Shih, "Interferometric Bell-state preparation using femtosecond-pulse-pumped spontaneous parametric down-conversion," Phys. Rev. A 63, 062301 (2001).
[CrossRef]

Kwiat, P. G.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773-R776 (1999).
[CrossRef]

D. V. Strekalov, T. B. Pittman, A. V. Sergienko, Y. H. Shih, and P. G. Kwiat, "Postselection-free energy-time entanglement," Phys. Rev. A 54, R1-R4 (1996).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New High-Intensity Source of Polarization-Entangled Photon Pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Kwon, O.

S.-Y. Baek, O. Kwon, and Y.-H. Kim, "High-Resolution Mode-Spacing Measurement of the Blue-Violet Diode Laser Using Interference of Felds Created with Time Delays Greater than the Coherence Time," Jpn. J. Appl. Phys. 46, 7720-7723 (2007).
[CrossRef]

Larchuk, T.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "Two-photon interference in a Mach-Zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Lu, Y. J.

Y. J. Lu, R. L. Campbell, and Z. Y. Ou, "Mode-locked two-photon states," Phys. Rev. Lett. 91, 163602 (2003).
[CrossRef] [PubMed]

Lundeen, J. S.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A.Walmsley, "Heralded Generation of Ultrafast Single Photons in Pure Quantum States," Phys. Rev. Lett. 100, 133601 (2008).
[CrossRef] [PubMed]

Mandel, L.

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical fourth-order interference," Phys. Rev. A 42, 2957-2965 (1990).
[CrossRef] [PubMed]

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef] [PubMed]

Mattle, K.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New High-Intensity Source of Polarization-Entangled Photon Pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

J. P. Dowling and G. J. Milburn, "Quantum technology: the second quantum revolution," Phil. Trans. R. Soc. Lond. A 361, 1655-1674 (2003).
[CrossRef]

Molina-Terriza, G.

A. Valencia, A. Cere, X. Shi, G. Molina-Terriza, and J. P. Torres, "Shaping theWaveform of Entangled Photons," Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

Monken, C. H.

M. A. Sagioro, C. Olindo, C. H. Monken, and S. Pádua, "Time control of two-photon interference," Phys. Rev. A 69, 053817 (2004).
[CrossRef]

Mosley, P. J.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A.Walmsley, "Heralded Generation of Ultrafast Single Photons in Pure Quantum States," Phys. Rev. Lett. 100, 133601 (2008).
[CrossRef] [PubMed]

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

Olindo, C.

M. A. Sagioro, C. Olindo, C. H. Monken, and S. Pádua, "Time control of two-photon interference," Phys. Rev. A 69, 053817 (2004).
[CrossRef]

Ou, Z. Y.

Y. J. Lu, R. L. Campbell, and Z. Y. Ou, "Mode-locked two-photon states," Phys. Rev. Lett. 91, 163602 (2003).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical fourth-order interference," Phys. Rev. A 42, 2957-2965 (1990).
[CrossRef] [PubMed]

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef] [PubMed]

Pádua, S.

M. A. Sagioro, C. Olindo, C. H. Monken, and S. Pádua, "Time control of two-photon interference," Phys. Rev. A 69, 053817 (2004).
[CrossRef]

Pittman, T. B.

D. V. Strekalov, T. B. Pittman, A. V. Sergienko, Y. H. Shih, and P. G. Kwiat, "Postselection-free energy-time entanglement," Phys. Rev. A 54, R1-R4 (1996).
[CrossRef] [PubMed]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

Rarity, J. G.

J. G. Rarity and P. R. Tapster, "Experimental Violation of Bell’s Inequality Based on Phase and Momentum," Phys. Rev. Lett. 64, 2495-2498 (1990).
[CrossRef] [PubMed]

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "Two-photon interference in a Mach-Zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Ribordy, G.

N. Gisin, G. Ribordy,W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Rubin, M. H.

Y.-H. Kim, M. V. Chekhova, S. P. Kulik, M. H. Rubin, and Y. Shih, "Interferometric Bell-state preparation using femtosecond-pulse-pumped spontaneous parametric down-conversion," Phys. Rev. A 63, 062301 (2001).
[CrossRef]

Sagioro, M. A.

M. A. Sagioro, C. Olindo, C. H. Monken, and S. Pádua, "Time control of two-photon interference," Phys. Rev. A 69, 053817 (2004).
[CrossRef]

Saleh, B. E. A.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "Two-photon interference in a Mach-Zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Sergienko, A. V.

D. V. Strekalov, T. B. Pittman, A. V. Sergienko, Y. H. Shih, and P. G. Kwiat, "Postselection-free energy-time entanglement," Phys. Rev. A 54, R1-R4 (1996).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New High-Intensity Source of Polarization-Entangled Photon Pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Shi, X.

A. Valencia, A. Cere, X. Shi, G. Molina-Terriza, and J. P. Torres, "Shaping theWaveform of Entangled Photons," Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

Shih, Y.

Y.-H. Kim, M. V. Chekhova, S. P. Kulik, M. H. Rubin, and Y. Shih, "Interferometric Bell-state preparation using femtosecond-pulse-pumped spontaneous parametric down-conversion," Phys. Rev. A 63, 062301 (2001).
[CrossRef]

Shih, Y. H.

Y.-H. Kim, V. Berardi, M. V. Chekhova, A. Garuccio, and Y. H. Shih, "Temporal indistinguishability and quantum interference," Phys. Rev. A 62, 43820 (2000).
[CrossRef]

D. V. Strekalov, T. B. Pittman, A. V. Sergienko, Y. H. Shih, and P. G. Kwiat, "Postselection-free energy-time entanglement," Phys. Rev. A 54, R1-R4 (1996).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New High-Intensity Source of Polarization-Entangled Photon Pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Y. H. Shih and C. O. Alley, "New Type of Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quants Produced by Optical Parametric Down Conversion," Phys. Rev. Lett. 61, 2921-2924 (1988).
[CrossRef] [PubMed]

Shimizu, R.

K. Edamatsu, R. Shimizu, and T. Itoh, "Measurement of the Photonic de BroglieWavelength of Entangled Photon Pairs Generated by Spontaneous Parametric Down-Conversion," Phys. Rev. Lett. 89, 213601 (2002).
[CrossRef] [PubMed]

Smith, B. J.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A.Walmsley, "Heralded Generation of Ultrafast Single Photons in Pure Quantum States," Phys. Rev. Lett. 100, 133601 (2008).
[CrossRef] [PubMed]

Strekalov, D. V.

D. V. Strekalov, T. B. Pittman, A. V. Sergienko, Y. H. Shih, and P. G. Kwiat, "Postselection-free energy-time entanglement," Phys. Rev. A 54, R1-R4 (1996).
[CrossRef] [PubMed]

Tapster, P. R.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "Two-photon interference in a Mach-Zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, "Experimental Violation of Bell’s Inequality Based on Phase and Momentum," Phys. Rev. Lett. 64, 2495-2498 (1990).
[CrossRef] [PubMed]

Teich, M. C.

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "Two-photon interference in a Mach-Zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Tittel, W.

N. Gisin, G. Ribordy,W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (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.

A. Valencia, A. Cere, X. Shi, G. Molina-Terriza, and J. P. Torres, "Shaping theWaveform of Entangled Photons," Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

Valencia, A.

A. Valencia, A. Cere, X. Shi, G. Molina-Terriza, and J. P. Torres, "Shaping theWaveform of Entangled Photons," Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

Viciani, S.

A. Zavatta, S. Viciani, and M. Bellini, "Recurrent fourth-order interference dips and peaks with a comblike two-photon entangled state," Phys. Rev. A 70, 023806 (2004).
[CrossRef]

Waks, E.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773-R776 (1999).
[CrossRef]

Wang, L. J.

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical fourth-order interference," Phys. Rev. A 42, 2957-2965 (1990).
[CrossRef] [PubMed]

Wasylczyk, P.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A.Walmsley, "Heralded Generation of Ultrafast Single Photons in Pure Quantum States," Phys. Rev. Lett. 100, 133601 (2008).
[CrossRef] [PubMed]

Weinfurter, H.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New High-Intensity Source of Polarization-Entangled Photon Pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

White, A. G.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773-R776 (1999).
[CrossRef]

Yamamoto, Y.

J. Jacobson, G. Bjork, I. Chuang, and Y. Yamamoto, "Photonic de Broglie waves," Phys. Rev. Lett. 74, 4835-4838 (1995).
[CrossRef] [PubMed]

Zavatta, A.

A. Zavatta, S. Viciani, and M. Bellini, "Recurrent fourth-order interference dips and peaks with a comblike two-photon entangled state," Phys. Rev. A 70, 023806 (2004).
[CrossRef]

Zbinden, H.

N. Gisin, G. Ribordy,W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (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.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New High-Intensity Source of Polarization-Entangled Photon Pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Zou, X. Y.

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical fourth-order interference," Phys. Rev. A 42, 2957-2965 (1990).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

Y.-H. Kim and W. P. Grice, "Generation of pulsed polarization-entangled two-photon state via temporal and spectral engineering," J. Mod. Opt. 49, 2309-2323 (2002).
[CrossRef]

Jpn. J. Appl. Phys. (1)

S.-Y. Baek, O. Kwon, and Y.-H. Kim, "High-Resolution Mode-Spacing Measurement of the Blue-Violet Diode Laser Using Interference of Felds Created with Time Delays Greater than the Coherence Time," Jpn. J. Appl. Phys. 46, 7720-7723 (2007).
[CrossRef]

Opt. Lett. (1)

Phil. Trans. R. Soc. Lond. A (1)

J. P. Dowling and G. J. Milburn, "Quantum technology: the second quantum revolution," Phil. Trans. R. Soc. Lond. A 361, 1655-1674 (2003).
[CrossRef]

Phys. Rev. A (10)

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, "Experiment on nonclassical fourth-order interference," Phys. Rev. A 42, 2957-2965 (1990).
[CrossRef] [PubMed]

Y.-H. Kim, V. Berardi, M. V. Chekhova, A. Garuccio, and Y. H. Shih, "Temporal indistinguishability and quantum interference," Phys. Rev. A 62, 43820 (2000).
[CrossRef]

S.-Y. Baek and Y.-H. Kim, "Spectral properties of entangled photon pairs generated via frequency-degenerate type-I spontaneous parametric down-conversion," Phys. Rev. A 77, 043807 (2008).
[CrossRef]

A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P Kulik, "Biphoton interference with a multimode pump," Phys. Rev. A 63, 053801 (2001).
[CrossRef]

M. A. Sagioro, C. Olindo, C. H. Monken, and S. Pádua, "Time control of two-photon interference," Phys. Rev. A 69, 053817 (2004).
[CrossRef]

A. Zavatta, S. Viciani, and M. Bellini, "Recurrent fourth-order interference dips and peaks with a comblike two-photon entangled state," Phys. Rev. A 70, 023806 (2004).
[CrossRef]

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773-R776 (1999).
[CrossRef]

Y.-H. Kim, M. V. Chekhova, S. P. Kulik, M. H. Rubin, and Y. Shih, "Interferometric Bell-state preparation using femtosecond-pulse-pumped spontaneous parametric down-conversion," Phys. Rev. A 63, 062301 (2001).
[CrossRef]

D. V. Strekalov, T. B. Pittman, A. V. Sergienko, Y. H. Shih, and P. G. Kwiat, "Postselection-free energy-time entanglement," Phys. Rev. A 54, R1-R4 (1996).
[CrossRef] [PubMed]

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, "Elimanating frequency and space-tie corelations in multiphoton states," Phys. Rev. A 64, 063815 (2001).
[CrossRef]

Phys. Rev. Lett. (11)

A. Valencia, A. Cere, X. Shi, G. Molina-Terriza, and J. P. Torres, "Shaping theWaveform of Entangled Photons," Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A.Walmsley, "Heralded Generation of Ultrafast Single Photons in Pure Quantum States," Phys. Rev. Lett. 100, 133601 (2008).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, "Experimental Violation of Bell’s Inequality Based on Phase and Momentum," Phys. Rev. Lett. 64, 2495-2498 (1990).
[CrossRef] [PubMed]

C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef] [PubMed]

J. Jacobson, G. Bjork, I. Chuang, and Y. Yamamoto, "Photonic de Broglie waves," Phys. Rev. Lett. 74, 4835-4838 (1995).
[CrossRef] [PubMed]

K. Edamatsu, R. Shimizu, and T. Itoh, "Measurement of the Photonic de BroglieWavelength of Entangled Photon Pairs Generated by Spontaneous Parametric Down-Conversion," Phys. Rev. Lett. 89, 213601 (2002).
[CrossRef] [PubMed]

J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, "Two-photon interference in a Mach-Zehnder interferometer," Phys. Rev. Lett. 65, 1348-1351 (1990).
[CrossRef] [PubMed]

Y. J. Lu, R. L. Campbell, and Z. Y. Ou, "Mode-locked two-photon states," Phys. Rev. Lett. 91, 163602 (2003).
[CrossRef] [PubMed]

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]

Y. H. Shih and C. O. Alley, "New Type of Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quants Produced by Optical Parametric Down Conversion," Phys. Rev. Lett. 61, 2921-2924 (1988).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New High-Intensity Source of Polarization-Entangled Photon Pairs," Phys. Rev. Lett. 75, 4337-4341 (1995).
[CrossRef] [PubMed]

Rev. Mod. Phys. (2)

N. Gisin, G. Ribordy,W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing with photonic qubits," Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

Other (4)

Y.-H. Kim, S. P. Kulik, M. V. Chekhova, W. P. Grice, and Y. Shih, "Experimental entanglement concentration and universal Bell-state synthesizer," Phys. Rev. A 67, 010301(R) (2003).
[CrossRef]

We have measured the transmission function of the interference filters used in this experiment with an Agilent 8453 UV/VIS spectro-photometer. We have found that the transmission function is indeed very close Gaussian centered at 810 nm as assumed in eq. (10).

The Wiener-Khinchine theorem states that the spectral power density of an optical field is related to its autocorrelation. They are, in fact, a Fourier transform pair. It is not difficult to show that the following relation holds, ⊗⌊FWHM = (4ln2/) 2 center/LFWHM. Here ⊗FWHM is the FWHM bandwidth of the field, center is the central wavelength of the laser (405 nm), and LFWHM is the FWHM width of the autocorrelation (interferogram) in Fig. 2.

J. W. Goodman, Statistical optics (Wiley, New York, 1985), p. 230.

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of the experiment. FPC is a fiber polarization controller and P1/P2 are trombone prisms. BS1/BS2 are 50:50 beam splitters. Note that x 1 and x 2 should be understood as path length differences.

Fig. 2.
Fig. 2.

Coherence property of the pump laser, studied with a Mach-Zehnder interferometer. Periodic recurrence of interference is observed. Inset shows the wave packet around x 2=0.

Fig. 3.
Fig. 3.

Coherence property of the pump laser, calculated with eq. (5). N=30 was used for evaluating the detection probability. Inset shows the wave packet around x 2=0.

Fig. 4.
Fig. 4.

The coincidence rate Rcd between single-photon detectors in modes c and d shows the Hong-Ou-Mandel dip with visibility better than 98%. No recurrence is observed. The single counting rates are roughly 43 kHz and 41 kHz for the single-photon detectors in mode c and in mode d, respectively.

Fig. 5.
Fig. 5.

The biphoton de Broglie wave packet measurement with different x1 values. (a) x 1=0 µm, (b) x 1=73 µm, (c) x 1=2834 µm, and (d) x 1=5668 µm. While the wave packet shapes are different, the modulation periods are the same in all cases at 405 nm. Note that the visibilities of the central wave packets are roughly the same at 98%.

Fig. 6.
Fig. 6.

Theoretical photonic de Broglie wave interference plotted using eq. (18) with N=30. The conditions are (a) x 1=0 µm, (b) x 1=73 µm, (c) x 1=2834 µm, and (d) x 1=5668 µm. The theoretical plots are in excellent agreement with the experimental data in Fig. 5.

Equations (23)

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

ρp= d ωp 𝓒 (ωp)ωpωp,
𝓒(ωp)=Σn=NN𝓒0(ωp)δ(ωpωp0nΔωp)Σn=NN𝓒0(ωp0+nΔωp),
𝓒0(ωp)exp ((ωpωp0)22δωp2) ,
Rf=tr [ρpEf()(t)Ef(+)(t)] ,
Rf=12+12𝓒p(τ2),
𝓒p(τ2)=Σn𝓢0(ωp0+nΔωp)cos((ωp0+nΔωp)τ2)Σn𝓢0(ωp0+nΔωp).
ψcd=(2c0d+0c2d)2,
ψ= d ωs d ωi δ (Δω)sinc(ΔkL2)eiΔkL2ωs,ωi,
ρ= d ωp 𝓢 (ωp) ψψ,
ϕ(ω)=exp ((ωω0)22Δω2)Δωπ,
Rcd d τ tr [ρEc()(t)Ed()(t+τ)Ed(+)(t+τ)Ec(+)(t)] ,
Rcd=1exp(Δω2τ122).
Ree tr [ρEe()(t)Ee()(t+τ)Ee(+)(t+τ)Ee(+)(t)] ,
ees (ωp)|dτ0Ee(+)(t+τ)Ee(+)(t)ψ2,
𝓡ees(ωp)=1+14exp((τ1τ2)2Δω22)
+14exp((τ1+τ2)2Δω22)12exp(τ22Δω22)
12cos(ωpτ2)×(1+exp(τ12Δω22)).
𝓡ees(τ1=0)=1 cos (ωpτ2).
Ree=Σn𝓢0(ωp)exp((ωp2ω0)22Δω2)𝓡ees(ωp)Σn𝓢0(ωp),
𝓡ee=1+14exp((τ1τ2)2Δω22)
+14exp((τ1+τ2)2Δω22)12exp(τ22Δω22)
12𝓒eff(τ2)(1+exp(τ12Δω22)),
𝓒eff(τ2)=Σn𝓢eff(ωp0+nΔωp)cos((ωp0+nΔωp)τ2)Σn𝓢eff(ωp0+nΔωp),

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