Abstract

We study spontaneous parametric down conversion (SPDC) in a one-dimensional photonic crystal designed to operate in a doubly resonant configuration, where the frequencies of the pump and the generated photons are both tuned to band-edge resonances. We investigate the spectral correlations of the generated photons as a function of the spectral width of the pump, and demonstrate that the SPDC generation rate can scale with the fifth power of the structure length in the limit of a quasi-continuous-wave pump. We show that such an unusual scaling can be simply connected with the scaling of second-harmonic generation in the same structure, illustrating the general link between spontaneous and stimulated parametric nonlinear processes.

© 2020 Optical Society of America

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References

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2017 (2)

P. R. Sharapova, K. H. Luo, H. Herrmann, M. Reichelt, T. Meier, and C. Silberhorn, New J. Phys. 19, 123009 (2017).
[Crossref]

X. Guo, C.-L. Zou, C. Schuck, H. Jung, R. Cheng, and H. X. Tang, Light Sci. Appl. 6, e16249 (2017).
[Crossref]

2014 (1)

C. H. Bennett and G. Brassard, Theor. Comput. Sci. 560, 7 (2014).
[Crossref]

2013 (1)

M. Liscidini and J. E. Sipe, Phys. Rev. Lett. 111, 193602 (2013).
[Crossref]

2012 (2)

M. Liscidini, L. G. Helt, and J. E. Sipe, Phys. Rev. A 85, 013833 (2012).
[Crossref]

L. G. Helt, M. Liscidini, and J. E. Sipe, J. Opt. Soc. Am. B 29, 2199 (2012).
[Crossref]

2010 (1)

Y. Dong and X. Zhang, Phys. Rev. A 81, 033806 (2010).
[Crossref]

2008 (2)

Z. Yang, M. Liscidini, and J. E. Sipe, Phys. Rev. A 77, 033808 (2008).
[Crossref]

J. P. Dowling, Contemp. Phys. 49, 125 (2008).
[Crossref]

2007 (5)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 (2007).
[Crossref]

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, Phys. Rev. Lett. 99, 053907 (2007).
[Crossref]

N. Mattiucci, G. D’Aguanno, M. Scalora, and M. J. Bloemer, J. Opt. Soc. Am. B 24, 877 (2007).
[Crossref]

Y. Zeng, Y. Fu, X. Chen, W. Lu, and H. Ågren, J. Opt. Soc. Am. B 24, 1365 (2007).
[Crossref]

M. Corona and A. B. U’Ren, Phys. Rev. A 76, 043829 (2007).
[Crossref]

2006 (1)

G. Weihs, Int. J. Mod. Phys. B 20, 1543 (2006).
[Crossref]

2005 (2)

M. Centini, J. Peřina, L. Sciscione, C. Sibilia, M. Scalora, M. J. Bloemer, and M. Bertolotti, Phys. Rev. A 72, 033806 (2005).
[Crossref]

W. T. M. Irvine, M. J. A. de Dood, and D. Bouwmeester, Phys. Rev. A 72, 043815 (2005).
[Crossref]

2004 (2)

F.-F. Ren, R. Li, C. Cheng, H.-T. Wang, J. Qiu, J. Si, and K. Hirao, Phys. Rev. B 70, 245109 (2004).
[Crossref]

A. N. Vamivakas, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. A 70, 043810 (2004).
[Crossref]

2003 (1)

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, Phys. Rev. E 68, 066617 (2003).
[Crossref]

2002 (2)

A. R. Cowan and J. F. Young, Phys. Rev. B 65, 085106 (2002).
[Crossref]

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, Phys. Rev. Lett. 89, 043901 (2002).
[Crossref]

2001 (1)

1999 (2)

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Phys. Rev. A 60, R773 (1999).
[Crossref]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[Crossref]

1997 (2)

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166 (1997).
[Crossref]

W. P. Grice and I. A. Walmsley, Phys. Rev. A 56, 1627 (1997).
[Crossref]

1996 (1)

Abram, I.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, Phys. Rev. Lett. 89, 043901 (2002).
[Crossref]

Ågren, H.

Aitchison, J. S.

Andreani, L. C.

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, Phys. Rev. Lett. 99, 053907 (2007).
[Crossref]

Angelis, C. D.

Appelbaum, I.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Phys. Rev. A 60, R773 (1999).
[Crossref]

Bennett, C. H.

C. H. Bennett and G. Brassard, Theor. Comput. Sci. 560, 7 (2014).
[Crossref]

Bertolotti, M.

M. Centini, J. Peřina, L. Sciscione, C. Sibilia, M. Scalora, M. J. Bloemer, and M. Bertolotti, Phys. Rev. A 72, 033806 (2005).
[Crossref]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[Crossref]

Bloemer, M. J.

N. Mattiucci, G. D’Aguanno, M. Scalora, and M. J. Bloemer, J. Opt. Soc. Am. B 24, 877 (2007).
[Crossref]

M. Centini, J. Peřina, L. Sciscione, C. Sibilia, M. Scalora, M. J. Bloemer, and M. Bertolotti, Phys. Rev. A 72, 033806 (2005).
[Crossref]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[Crossref]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166 (1997).
[Crossref]

Bouwmeester, D.

W. T. M. Irvine, M. J. A. de Dood, and D. Bouwmeester, Phys. Rev. A 72, 043815 (2005).
[Crossref]

Bowden, C. M.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[Crossref]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166 (1997).
[Crossref]

Brassard, G.

C. H. Bennett and G. Brassard, Theor. Comput. Sci. 560, 7 (2014).
[Crossref]

Centini, M.

M. Centini, J. Peřina, L. Sciscione, C. Sibilia, M. Scalora, M. J. Bloemer, and M. Bertolotti, Phys. Rev. A 72, 033806 (2005).
[Crossref]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[Crossref]

Chen, X.

Cheng, C.

F.-F. Ren, R. Li, C. Cheng, H.-T. Wang, J. Qiu, J. Si, and K. Hirao, Phys. Rev. B 70, 245109 (2004).
[Crossref]

Cheng, R.

X. Guo, C.-L. Zou, C. Schuck, H. Jung, R. Cheng, and H. X. Tang, Light Sci. Appl. 6, e16249 (2017).
[Crossref]

Corona, M.

M. Corona and A. B. U’Ren, Phys. Rev. A 76, 043829 (2007).
[Crossref]

Cowan, A. R.

A. R. Cowan and J. F. Young, Phys. Rev. B 65, 085106 (2002).
[Crossref]

D’Aguanno, G.

N. Mattiucci, G. D’Aguanno, M. Scalora, and M. J. Bloemer, J. Opt. Soc. Am. B 24, 877 (2007).
[Crossref]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[Crossref]

De Angelis, C.

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, Phys. Rev. Lett. 99, 053907 (2007).
[Crossref]

de Dood, M. J. A.

W. T. M. Irvine, M. J. A. de Dood, and D. Bouwmeester, Phys. Rev. A 72, 043815 (2005).
[Crossref]

de Sterke, C. M.

Dong, Y.

Y. Dong and X. Zhang, Phys. Rev. A 81, 033806 (2010).
[Crossref]

Dowling, J. P.

J. P. Dowling, Contemp. Phys. 49, 125 (2008).
[Crossref]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 (2007).
[Crossref]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166 (1997).
[Crossref]

Dumeige, Y.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, Phys. Rev. E 68, 066617 (2003).
[Crossref]

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, Phys. Rev. Lett. 89, 043901 (2002).
[Crossref]

Eberhard, P. H.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Phys. Rev. A 60, R773 (1999).
[Crossref]

Fu, Y.

Grice, W. P.

W. P. Grice and I. A. Walmsley, Phys. Rev. A 56, 1627 (1997).
[Crossref]

Gringoli, F.

Guo, X.

X. Guo, C.-L. Zou, C. Schuck, H. Jung, R. Cheng, and H. X. Tang, Light Sci. Appl. 6, e16249 (2017).
[Crossref]

Haus, J. W.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166 (1997).
[Crossref]

Helt, L. G.

L. G. Helt, M. Liscidini, and J. E. Sipe, J. Opt. Soc. Am. B 29, 2199 (2012).
[Crossref]

M. Liscidini, L. G. Helt, and J. E. Sipe, Phys. Rev. A 85, 013833 (2012).
[Crossref]

Herrmann, H.

P. R. Sharapova, K. H. Luo, H. Herrmann, M. Reichelt, T. Meier, and C. Silberhorn, New J. Phys. 19, 123009 (2017).
[Crossref]

Hirao, K.

F.-F. Ren, R. Li, C. Cheng, H.-T. Wang, J. Qiu, J. Si, and K. Hirao, Phys. Rev. B 70, 245109 (2004).
[Crossref]

Irvine, W. T. M.

W. T. M. Irvine, M. J. A. de Dood, and D. Bouwmeester, Phys. Rev. A 72, 043815 (2005).
[Crossref]

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Jung, H.

X. Guo, C.-L. Zou, C. Schuck, H. Jung, R. Cheng, and H. X. Tang, Light Sci. Appl. 6, e16249 (2017).
[Crossref]

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 (2007).
[Crossref]

Kwiat, P. G.

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Phys. Rev. A 60, R773 (1999).
[Crossref]

Letartre, X.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, Phys. Rev. E 68, 066617 (2003).
[Crossref]

Levenson, A.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, Phys. Rev. E 68, 066617 (2003).
[Crossref]

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, Phys. Rev. Lett. 89, 043901 (2002).
[Crossref]

Li, R.

F.-F. Ren, R. Li, C. Cheng, H.-T. Wang, J. Qiu, J. Si, and K. Hirao, Phys. Rev. B 70, 245109 (2004).
[Crossref]

Liscidini, M.

M. Liscidini and J. E. Sipe, Phys. Rev. Lett. 111, 193602 (2013).
[Crossref]

M. Liscidini, L. G. Helt, and J. E. Sipe, Phys. Rev. A 85, 013833 (2012).
[Crossref]

L. G. Helt, M. Liscidini, and J. E. Sipe, J. Opt. Soc. Am. B 29, 2199 (2012).
[Crossref]

Z. Yang, M. Liscidini, and J. E. Sipe, Phys. Rev. A 77, 033808 (2008).
[Crossref]

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, Phys. Rev. Lett. 99, 053907 (2007).
[Crossref]

Locatelli, A.

M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, Phys. Rev. Lett. 99, 053907 (2007).
[Crossref]

Lu, W.

Luo, K. H.

P. R. Sharapova, K. H. Luo, H. Herrmann, M. Reichelt, T. Meier, and C. Silberhorn, New J. Phys. 19, 123009 (2017).
[Crossref]

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166 (1997).
[Crossref]

Mattiucci, N.

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Meier, T.

P. R. Sharapova, K. H. Luo, H. Herrmann, M. Reichelt, T. Meier, and C. Silberhorn, New J. Phys. 19, 123009 (2017).
[Crossref]

Mériadec, C.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, Phys. Rev. Lett. 89, 043901 (2002).
[Crossref]

Midrio, M.

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 (2007).
[Crossref]

Modotto, D.

Monnier, P.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, Phys. Rev. Lett. 89, 043901 (2002).
[Crossref]

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 (2007).
[Crossref]

Nalesso, G. F.

Nefedov, I.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[Crossref]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 (2007).
[Crossref]

Palik, E. D.

E. D. Palik and E. J. Prucha, Handbook of Optical Constants of Solids (Academic, 1997).

Perina, J.

M. Centini, J. Peřina, L. Sciscione, C. Sibilia, M. Scalora, M. J. Bloemer, and M. Bertolotti, Phys. Rev. A 72, 033806 (2005).
[Crossref]

Prucha, E. J.

E. D. Palik and E. J. Prucha, Handbook of Optical Constants of Solids (Academic, 1997).

Qiu, J.

F.-F. Ren, R. Li, C. Cheng, H.-T. Wang, J. Qiu, J. Si, and K. Hirao, Phys. Rev. B 70, 245109 (2004).
[Crossref]

Raineri, F.

Y. Dumeige, F. Raineri, A. Levenson, and X. Letartre, Phys. Rev. E 68, 066617 (2003).
[Crossref]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, Rev. Mod. Phys. 79, 135 (2007).
[Crossref]

Reichelt, M.

P. R. Sharapova, K. H. Luo, H. Herrmann, M. Reichelt, T. Meier, and C. Silberhorn, New J. Phys. 19, 123009 (2017).
[Crossref]

Ren, F.-F.

F.-F. Ren, R. Li, C. Cheng, H.-T. Wang, J. Qiu, J. Si, and K. Hirao, Phys. Rev. B 70, 245109 (2004).
[Crossref]

Sagnes, I.

Y. Dumeige, I. Sagnes, P. Monnier, P. Vidakovic, I. Abram, C. Mériadec, and A. Levenson, Phys. Rev. Lett. 89, 043901 (2002).
[Crossref]

Saleh, B. E. A.

A. N. Vamivakas, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. A 70, 043810 (2004).
[Crossref]

Scalora, M.

N. Mattiucci, G. D’Aguanno, M. Scalora, and M. J. Bloemer, J. Opt. Soc. Am. B 24, 877 (2007).
[Crossref]

M. Centini, J. Peřina, L. Sciscione, C. Sibilia, M. Scalora, M. J. Bloemer, and M. Bertolotti, Phys. Rev. A 72, 033806 (2005).
[Crossref]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[Crossref]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166 (1997).
[Crossref]

Schuck, C.

X. Guo, C.-L. Zou, C. Schuck, H. Jung, R. Cheng, and H. X. Tang, Light Sci. Appl. 6, e16249 (2017).
[Crossref]

Sciscione, L.

M. Centini, J. Peřina, L. Sciscione, C. Sibilia, M. Scalora, M. J. Bloemer, and M. Bertolotti, Phys. Rev. A 72, 033806 (2005).
[Crossref]

Sergienko, A. V.

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Appl. Opt. (1)

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J. P. Dowling, Contemp. Phys. 49, 125 (2008).
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Int. J. Mod. Phys. B (1)

G. Weihs, Int. J. Mod. Phys. B 20, 1543 (2006).
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Phys. Rev. A (10)

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Phys. Rev. A 60, R773 (1999).
[Crossref]

M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, Phys. Rev. A 56, 3166 (1997).
[Crossref]

A. N. Vamivakas, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. A 70, 043810 (2004).
[Crossref]

M. Centini, J. Peřina, L. Sciscione, C. Sibilia, M. Scalora, M. J. Bloemer, and M. Bertolotti, Phys. Rev. A 72, 033806 (2005).
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Z. Yang, M. Liscidini, and J. E. Sipe, Phys. Rev. A 77, 033808 (2008).
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M. Liscidini, L. G. Helt, and J. E. Sipe, Phys. Rev. A 85, 013833 (2012).
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W. P. Grice and I. A. Walmsley, Phys. Rev. A 56, 1627 (1997).
[Crossref]

M. Corona and A. B. U’Ren, Phys. Rev. A 76, 043829 (2007).
[Crossref]

Y. Dong and X. Zhang, Phys. Rev. A 81, 033806 (2010).
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A. R. Cowan and J. F. Young, Phys. Rev. B 65, 085106 (2002).
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M. Liscidini, A. Locatelli, L. C. Andreani, and C. De Angelis, Phys. Rev. Lett. 99, 053907 (2007).
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M. Liscidini and J. E. Sipe, Phys. Rev. Lett. 111, 193602 (2013).
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Other (3)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

E. D. Palik and E. J. Prucha, Handbook of Optical Constants of Solids (Academic, 1997).

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, The Oxford Series in Electrical and Computer Engineering (Oxford University, 2006).

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Figures (4)

Fig. 1.
Fig. 1. (a) Sketch of the structure and process under consideration; transmission spectra of the structure in the pump (b) and second-harmonic (c) energy range. We consider $ N = 30 $ (solid line) and $ N = 50 $ (dashed line) periods.
Fig. 2.
Fig. 2. (a) JSI for $ N = 30 $ periods, in the case of doubly resonant SPDC for a short ($ {\sigma _E} = 10\,\,\text{meV} $) pulse; (b) resonant conditions and PBGs superposed on the JSI: dashed red lines correspond to resonant enhancement of the fundamental field [see Fig. 1(c)]; solid green lines correspond to resonant enhancement of the second-harmonic field [see Fig. 1(b)]; dashed regions indicated with $ {\text{PBG}_{\text{F}}} $ and $ {\text{PBG}_{\text{S}}} $ correspond to PBG frequencies for the fundamental and second-harmonic fields, respectively [see Figs. 1(b) and 1(c)].
Fig. 3.
Fig. 3. JSI for $ N = 30 $ periods, in the case of doubly resonant SPDC for a long ($ {\sigma _E} = 1\,\,\text{meV} $) pump pulse tuned at 1.24 eV.
Fig. 4.
Fig. 4. Generation rate (per pump intensity) and resonance half width versus number of periods $ N $ for a cw pump at 1.24 eV. We assume $ {\bar \chi _2} = 100\;\text{pm/V} $.

Equations (4)

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| II = 1 2 d ω 1 d ω 2 ϕ ( ω 1 , ω 2 ) a ω 1 a ω 2 | vac ,
ϕ ( ω 1 , ω 2 ) = i α χ ¯ 2 β ω F 2 ω S 8 π ε 0 n F 2 n S c 3 A × ϕ P ( ω 1 + ω 2 ) J ( ω 1 + ω 2 , ω 1 , ω 2 )
J ( ω 1 , ω 2 , ω 3 ) = O L d z χ 2 i j k ( z ) χ ¯ 2 f R , ω 1 i ( z ) f R , ω 2 j ( z ) f L , ω 1 + ω 2 k ( z )
| β | 2 = | α | 2 χ ¯ 2 2 ω F 2 ω S 8 π ε 0 n F 2 n S c 3 A × d ω 1 d ω 2 | ϕ P ( ω 1 + ω 2 ) J ( ω 1 + ω 2 , ω 1 , ω 2 ) | 2 ,

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