Abstract

We study theoretically the Čerenkov-type second-harmonic generation by a Gaussian beam in quadratic nonlinear media with periodically reverted ferroelectric domains. In particular, we consider the role of fundamental beam width on the harmonic emission process. We show that varying the beam width of the fundamental wave not only affects the strength of the emitted Čerenkov harmonic signal but also changes the sensitivity of the emission process to wavelength tuning.

© 2012 Optical Society of America

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  1. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [CrossRef]
  2. R. C. Miller, “Optical haromonic generation in single crystal BaTiO3,” Phys. Rev. 134, A1313–A1319 (1964).
    [CrossRef]
  3. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, 1991).
  4. J. A. Amstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  5. S. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
    [CrossRef]
  6. V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
    [CrossRef]
  7. N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
    [CrossRef]
  8. A. Arie and N. Voloch-Bloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photon. Rev. 4, 355–373 (2010).
    [CrossRef]
  9. Y. Sheng, J. Dou, B. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 011101 (2007).
    [CrossRef]
  10. S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, “Čerenkov-type second harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45, 1465–1472 (2009).
    [CrossRef]
  11. Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov radiation in nonlinear photonic crystal waveguides,” Phys. Rev. Lett. 100, 163904 (2008).
    [CrossRef]
  12. Y. Sheng, S. M. Saltiel, W. Krolikowski, A. Arie, K. Koynov, and Y. S. Kivshar, “Čerenkov-type second-harmonic generation with fundamental beams of different polarizations,” Opt. Lett. 35, 1317–1319 (2010).
    [CrossRef]
  13. W. Wang, Y. Sheng, Y. Kong, A. Arie, and W. Krolikowski, “Multiple Čerenkov second-harmonic waves in a two-dimensional nonlinear photonic structure,” Opt. Lett. 35, 3790–3792 (2010).
    [CrossRef]
  14. I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulse in the Laue scheme,” Opt. Spectrosc. 105, 79–84 (2008).
    [CrossRef]
  15. P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, “Directional dependence of the second harmonic response in two-dimensional nonlinear photonic crystals,” Appl. Phys. Lett. 96, 261111 (2010).
    [CrossRef]
  16. J. V. Jelley, Cerenkov Radiation (Pergamon, 1958).
  17. A. Zembrod, H. Puell, and J. Giordmaine, “Surface radiation from non-linear optical polarisation,” Opt. Quantum Electron. 1, 64–66 (1969).
    [CrossRef]
  18. A. A. Kaminskii, H. Nishioka, K. Ueda, W. Odajima, M. Tateno, K. Sasaki, and A. V. Butashin, “Second-harmonic generation with Cherenkov-type phase matching in a bulk nonlinear LaBGeO5 crystal,” Quantum Electron. 26, 381–382 (1996).
    [CrossRef]
  19. V. Vacaitis, “Cherenkov-type phase matching in bulk KDP crystal,” Opt. Commun. 209, 485–490 (2002).
    [CrossRef]
  20. Y. Sheng, A. Best, H. Butt, W. Krolikowski, A. Arie, and K. Koynov, “Three-dimensional ferroelectric domain visualization by Čerenkov-type second harmonic generation,” Opt. Express 18, 16539–16545 (2010).
    [CrossRef]
  21. A. Fragemann, V. Pasiskevicus, and F. Laurell, “Second-order nonlinearites in the domain walls of periodically poled KTiOPO4,” Appl. Phys. Lett. 85, 375–377(2004).
    [CrossRef]
  22. X. Deng and X. Chen, “Domain wall characterization in ferroelectrics by using localized nonlinearities,” Opt. Express 18, 15597–15602 (2010).
    [CrossRef]
  23. A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361–375 (2007).
    [CrossRef]
  24. A. Ganany, A. Arie, and S. M. Saltiel, “Quasi-phase matching in LiNbO3 using nonlinear coefficients in the XY plane,” Appl. Phys. B 85, 97–100 (2006).
    [CrossRef]
  25. G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–375 (1984).
    [CrossRef]

2010 (6)

2009 (1)

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, “Čerenkov-type second harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45, 1465–1472 (2009).
[CrossRef]

2008 (2)

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov radiation in nonlinear photonic crystal waveguides,” Phys. Rev. Lett. 100, 163904 (2008).
[CrossRef]

I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulse in the Laue scheme,” Opt. Spectrosc. 105, 79–84 (2008).
[CrossRef]

2007 (2)

Y. Sheng, J. Dou, B. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 011101 (2007).
[CrossRef]

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361–375 (2007).
[CrossRef]

2006 (1)

A. Ganany, A. Arie, and S. M. Saltiel, “Quasi-phase matching in LiNbO3 using nonlinear coefficients in the XY plane,” Appl. Phys. B 85, 97–100 (2006).
[CrossRef]

2004 (1)

A. Fragemann, V. Pasiskevicus, and F. Laurell, “Second-order nonlinearites in the domain walls of periodically poled KTiOPO4,” Appl. Phys. Lett. 85, 375–377(2004).
[CrossRef]

2002 (1)

V. Vacaitis, “Cherenkov-type phase matching in bulk KDP crystal,” Opt. Commun. 209, 485–490 (2002).
[CrossRef]

2000 (1)

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[CrossRef]

1998 (1)

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[CrossRef]

1997 (1)

S. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

1996 (1)

A. A. Kaminskii, H. Nishioka, K. Ueda, W. Odajima, M. Tateno, K. Sasaki, and A. V. Butashin, “Second-harmonic generation with Cherenkov-type phase matching in a bulk nonlinear LaBGeO5 crystal,” Quantum Electron. 26, 381–382 (1996).
[CrossRef]

1984 (1)

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–375 (1984).
[CrossRef]

1969 (1)

A. Zembrod, H. Puell, and J. Giordmaine, “Surface radiation from non-linear optical polarisation,” Opt. Quantum Electron. 1, 64–66 (1969).
[CrossRef]

1964 (1)

R. C. Miller, “Optical haromonic generation in single crystal BaTiO3,” Phys. Rev. 134, A1313–A1319 (1964).
[CrossRef]

1962 (1)

J. A. Amstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Amstrong, J. A.

J. A. Amstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Arie, A.

A. Arie and N. Voloch-Bloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photon. Rev. 4, 355–373 (2010).
[CrossRef]

Y. Sheng, S. M. Saltiel, W. Krolikowski, A. Arie, K. Koynov, and Y. S. Kivshar, “Čerenkov-type second-harmonic generation with fundamental beams of different polarizations,” Opt. Lett. 35, 1317–1319 (2010).
[CrossRef]

W. Wang, Y. Sheng, Y. Kong, A. Arie, and W. Krolikowski, “Multiple Čerenkov second-harmonic waves in a two-dimensional nonlinear photonic structure,” Opt. Lett. 35, 3790–3792 (2010).
[CrossRef]

Y. Sheng, A. Best, H. Butt, W. Krolikowski, A. Arie, and K. Koynov, “Three-dimensional ferroelectric domain visualization by Čerenkov-type second harmonic generation,” Opt. Express 18, 16539–16545 (2010).
[CrossRef]

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, “Čerenkov-type second harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45, 1465–1472 (2009).
[CrossRef]

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361–375 (2007).
[CrossRef]

A. Ganany, A. Arie, and S. M. Saltiel, “Quasi-phase matching in LiNbO3 using nonlinear coefficients in the XY plane,” Appl. Phys. B 85, 97–100 (2006).
[CrossRef]

Bahabad, A.

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361–375 (2007).
[CrossRef]

Bausa, L. E.

P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, “Directional dependence of the second harmonic response in two-dimensional nonlinear photonic crystals,” Appl. Phys. Lett. 96, 261111 (2010).
[CrossRef]

Berger, V.

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[CrossRef]

Best, A.

Bloembergen, N.

J. A. Amstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Broderick, N. G. R.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[CrossRef]

Butashin, A. V.

A. A. Kaminskii, H. Nishioka, K. Ueda, W. Odajima, M. Tateno, K. Sasaki, and A. V. Butashin, “Second-harmonic generation with Cherenkov-type phase matching in a bulk nonlinear LaBGeO5 crystal,” Quantum Electron. 26, 381–382 (1996).
[CrossRef]

Butt, H.

Chen, X.

Cheng, B.

Y. Sheng, J. Dou, B. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 011101 (2007).
[CrossRef]

Chirkin, A. S.

I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulse in the Laue scheme,” Opt. Spectrosc. 105, 79–84 (2008).
[CrossRef]

Deng, X.

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, 1991).

Dou, J.

Y. Sheng, J. Dou, B. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 011101 (2007).
[CrossRef]

Ducuing, J.

J. A. Amstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Edwards, G. J.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–375 (1984).
[CrossRef]

Fragemann, A.

A. Fragemann, V. Pasiskevicus, and F. Laurell, “Second-order nonlinearites in the domain walls of periodically poled KTiOPO4,” Appl. Phys. Lett. 85, 375–377(2004).
[CrossRef]

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Ganany, A.

A. Ganany, A. Arie, and S. M. Saltiel, “Quasi-phase matching in LiNbO3 using nonlinear coefficients in the XY plane,” Appl. Phys. B 85, 97–100 (2006).
[CrossRef]

Gao, Z. D.

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov radiation in nonlinear photonic crystal waveguides,” Phys. Rev. Lett. 100, 163904 (2008).
[CrossRef]

Garcia, B. J.

P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, “Directional dependence of the second harmonic response in two-dimensional nonlinear photonic crystals,” Appl. Phys. Lett. 96, 261111 (2010).
[CrossRef]

Giordmaine, J.

A. Zembrod, H. Puell, and J. Giordmaine, “Surface radiation from non-linear optical polarisation,” Opt. Quantum Electron. 1, 64–66 (1969).
[CrossRef]

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, 1991).

Habshoosh, N.

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quantum Electron. 39, 361–375 (2007).
[CrossRef]

Hanna, D. C.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[CrossRef]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Jelley, J. V.

J. V. Jelley, Cerenkov Radiation (Pergamon, 1958).

Kaminskii, A. A.

A. A. Kaminskii, H. Nishioka, K. Ueda, W. Odajima, M. Tateno, K. Sasaki, and A. V. Butashin, “Second-harmonic generation with Cherenkov-type phase matching in a bulk nonlinear LaBGeO5 crystal,” Quantum Electron. 26, 381–382 (1996).
[CrossRef]

Kivshar, Y. S.

Y. Sheng, S. M. Saltiel, W. Krolikowski, A. Arie, K. Koynov, and Y. S. Kivshar, “Čerenkov-type second-harmonic generation with fundamental beams of different polarizations,” Opt. Lett. 35, 1317–1319 (2010).
[CrossRef]

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, “Čerenkov-type second harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45, 1465–1472 (2009).
[CrossRef]

Kong, Y.

Koynov, K.

Koyonov, K.

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, “Čerenkov-type second harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45, 1465–1472 (2009).
[CrossRef]

Krolikowski, W.

Laurell, F.

A. Fragemann, V. Pasiskevicus, and F. Laurell, “Second-order nonlinearites in the domain walls of periodically poled KTiOPO4,” Appl. Phys. Lett. 85, 375–377(2004).
[CrossRef]

Lawrence, M.

G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373–375 (1984).
[CrossRef]

Ma, B.

Y. Sheng, J. Dou, B. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 011101 (2007).
[CrossRef]

Miller, R. C.

R. C. Miller, “Optical haromonic generation in single crystal BaTiO3,” Phys. Rev. 134, A1313–A1319 (1964).
[CrossRef]

Ming, N. B.

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov radiation in nonlinear photonic crystal waveguides,” Phys. Rev. Lett. 100, 163904 (2008).
[CrossRef]

S. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

Molina, P.

P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, “Directional dependence of the second harmonic response in two-dimensional nonlinear photonic crystals,” Appl. Phys. Lett. 96, 261111 (2010).
[CrossRef]

Neshev, D. N.

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, “Čerenkov-type second harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45, 1465–1472 (2009).
[CrossRef]

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, 1991).

Nishioka, H.

A. A. Kaminskii, H. Nishioka, K. Ueda, W. Odajima, M. Tateno, K. Sasaki, and A. V. Butashin, “Second-harmonic generation with Cherenkov-type phase matching in a bulk nonlinear LaBGeO5 crystal,” Quantum Electron. 26, 381–382 (1996).
[CrossRef]

Odajima, W.

A. A. Kaminskii, H. Nishioka, K. Ueda, W. Odajima, M. Tateno, K. Sasaki, and A. V. Butashin, “Second-harmonic generation with Cherenkov-type phase matching in a bulk nonlinear LaBGeO5 crystal,” Quantum Electron. 26, 381–382 (1996).
[CrossRef]

Offerhaus, H. L.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[CrossRef]

Ozheredov, I. A.

I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulse in the Laue scheme,” Opt. Spectrosc. 105, 79–84 (2008).
[CrossRef]

Pasiskevicus, V.

A. Fragemann, V. Pasiskevicus, and F. Laurell, “Second-order nonlinearites in the domain walls of periodically poled KTiOPO4,” Appl. Phys. Lett. 85, 375–377(2004).
[CrossRef]

Pershan, P. S.

J. A. Amstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Puell, H.

A. Zembrod, H. Puell, and J. Giordmaine, “Surface radiation from non-linear optical polarisation,” Opt. Quantum Electron. 1, 64–66 (1969).
[CrossRef]

Qi, Z.

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov radiation in nonlinear photonic crystal waveguides,” Phys. Rev. Lett. 100, 163904 (2008).
[CrossRef]

Ramirez, M. O.

P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, “Directional dependence of the second harmonic response in two-dimensional nonlinear photonic crystals,” Appl. Phys. Lett. 96, 261111 (2010).
[CrossRef]

Richardson, D. J.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[CrossRef]

Ross, G. W.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[CrossRef]

Saltiel, S. M.

Y. Sheng, S. M. Saltiel, W. Krolikowski, A. Arie, K. Koynov, and Y. S. Kivshar, “Čerenkov-type second-harmonic generation with fundamental beams of different polarizations,” Opt. Lett. 35, 1317–1319 (2010).
[CrossRef]

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, “Čerenkov-type second harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45, 1465–1472 (2009).
[CrossRef]

A. Ganany, A. Arie, and S. M. Saltiel, “Quasi-phase matching in LiNbO3 using nonlinear coefficients in the XY plane,” Appl. Phys. B 85, 97–100 (2006).
[CrossRef]

Sasaki, K.

A. A. Kaminskii, H. Nishioka, K. Ueda, W. Odajima, M. Tateno, K. Sasaki, and A. V. Butashin, “Second-harmonic generation with Cherenkov-type phase matching in a bulk nonlinear LaBGeO5 crystal,” Quantum Electron. 26, 381–382 (1996).
[CrossRef]

Sheng, Y.

Shumitski, A. V.

I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulse in the Laue scheme,” Opt. Spectrosc. 105, 79–84 (2008).
[CrossRef]

Shutov, I. V.

I. V. Shutov, I. A. Ozheredov, A. V. Shumitski, and A. S. Chirkin, “Second harmonic generation by femtosecond laser pulse in the Laue scheme,” Opt. Spectrosc. 105, 79–84 (2008).
[CrossRef]

Tateno, M.

A. A. Kaminskii, H. Nishioka, K. Ueda, W. Odajima, M. Tateno, K. Sasaki, and A. V. Butashin, “Second-harmonic generation with Cherenkov-type phase matching in a bulk nonlinear LaBGeO5 crystal,” Quantum Electron. 26, 381–382 (1996).
[CrossRef]

Ueda, K.

A. A. Kaminskii, H. Nishioka, K. Ueda, W. Odajima, M. Tateno, K. Sasaki, and A. V. Butashin, “Second-harmonic generation with Cherenkov-type phase matching in a bulk nonlinear LaBGeO5 crystal,” Quantum Electron. 26, 381–382 (1996).
[CrossRef]

Vacaitis, V.

V. Vacaitis, “Cherenkov-type phase matching in bulk KDP crystal,” Opt. Commun. 209, 485–490 (2002).
[CrossRef]

Voloch-Bloch, N.

A. Arie and N. Voloch-Bloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photon. Rev. 4, 355–373 (2010).
[CrossRef]

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, “Čerenkov-type second harmonic generation in two-dimensional nonlinear photonic structures,” IEEE J. Quantum Electron. 45, 1465–1472 (2009).
[CrossRef]

Wang, W.

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Zembrod, A.

A. Zembrod, H. Puell, and J. Giordmaine, “Surface radiation from non-linear optical polarisation,” Opt. Quantum Electron. 1, 64–66 (1969).
[CrossRef]

Zhang, D.

Y. Sheng, J. Dou, B. Ma, B. Cheng, and D. Zhang, “Broadband efficient second harmonic generation in media with a short-range order,” Appl. Phys. Lett. 91, 011101 (2007).
[CrossRef]

Zhang, Y.

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, “Nonlinear Čerenkov radiation in nonlinear photonic crystal waveguides,” Phys. Rev. Lett. 100, 163904 (2008).
[CrossRef]

Zhu, S.

S. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

(a) Traditional quasi-phase-matched SHG with fundamental beam propagating along normal to the domain walls ( x axis). (b) Čerenkov-type SHG with fundamental wave directed along the domain walls ( z axis). (c) Triangle constructed by the vectors k 2 , 2 k 1 , and κ x when the Čerenkov phase-matching condition is fulfilled, i.e., when the argument of the sinc function in Eq. (8) equals to zero. (d) Domain pattern described by Eq. (6). Unless otherwise specified, the pump Gaussian beam is located centrally at the position x 0 = D Λ / 2 .

Fig. 2.
Fig. 2.

(a) Angular distribution of Čerenkov SHG signal for the fundamental wavelength of 1.108 μm. (b) Typical dependence of the intensity of the Čerenkov SH on the propagation length ( κ x = 6.984 μm 1 ).

Fig. 3.
Fig. 3.

Left column: spectral response of the Čerenkov SHG for different beam widths. Right column: spatial Fourier spectrum of the Čerenkov emission. From top to bottom, the beam widths of the fundamental wave are 60, 10, 5, and 2 μm. The plots are normalized to their individual maximum value. Vertical green and red lines indicate two particular wavelengths of the fundamental wave, λ 1 = 1.038 μm and λ 1 = 1.108 μm , respectively.

Fig. 4.
Fig. 4.

Čerenkov SH signal versus beam width for different wavelengths of the fundamental beam. (a)  λ = 1.108 μm and (b)  λ = 1.038 μm with the peak intensity of the fundamental wave being kept constant in both cases. (c)  λ = 1.108 μm and (d)  λ = 1.038 μm with constant pump power of the fundamental wave. κ x = 6.984 μm 1 for λ = 1.108 μm and κ x = 8.016 μm 1 for λ = 1.038 μm .

Fig. 5.
Fig. 5.

Čerenkov SH emission calculated when the fundamental beam is located at different positions ( λ = 1.108 μm and κ x = 6.984 μm 1 ). The blocks in gray and white represent the positive and reversed domains, respectively. It is clear that, in the case of a narrow beam, the Čerenkov SH is stronger when the beam is located exactly at the domain wall.

Equations (9)

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E 2 ω ( x , z , t ) = A 2 ( x , z ) e i ( 2 ω t k 2 · z ) .
( / z + i 2 k 2 2 / x 2 ) A 2 ( x , z ) = i g ( x ) β 2 I 1 F 1 2 ( x ) e i Δ k z ,
A 2 ( κ x , z ) = A 2 ( x , z ) e i κ x x d x .
( / z i κ x 2 / 2 k 2 ) A 2 ( κ x , z ) = Γ g ( x ) F 1 2 ( x ) e i κ x x e i Δ k z d x ,
A 2 ( κ x , z ) = Γ z e i z ( Δ k + κ x 2 / 2 k 2 ) / 2 × sinc [ z ( Δ k κ x 2 / 2 k 2 ) / 2 ] g ( x ) F 1 2 ( x ) e i κ x x d x ,
g ( x ) = m = 0 , ± 1 , ± 2 , g m e i m G 0 x .
g ( x ) F 1 2 ( x ) e i κ x x d x = a ( π / 2 ) 1 / 2 × m = 0 , ± 1 , ± 2 , g m e a 2 ( m G 0 + κ x ) 2 / 8 e i ( m G 0 + κ x ) x 0 .
S 2 ( κ x , z ) = π a 2 z 2 Γ 2 / 2 × { sinc [ z ( Δ k κ x 2 / 2 k 2 ) / 2 ] } 2 × ( m = 0 , ± 1 , ± 2 , g m e a 2 ( m G 0 + κ x ) 2 / 8 e i m x 0 G 0 ) 2 .
Δ k κ x 2 / 2 k 2 = 0 ,

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