Abstract

We study the second-harmonic generation in quadratic nonlinear media with localized spatial modulation of χ(2) response. We demonstrate that the emission of Čerenkov second-harmonic takes place only when the fundamental beam illuminates the region of χ(2) variation. This proves that the sharp modulation of the χ(2) nonlinearity constitutes a sufficient condition for the emission of Čerenkov second harmonic in bulk materials. Our calculations are in excellent agreement with simple analytical approach utilizing the concept of reciprocal vectors representing the Fourier spectrum of the modulation of χ(2).

© 2012 Optical Society of America

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References

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  1. E. Mathieu, ZAMP 20, 433 (1969).
    [CrossRef]
  2. S. M. Saltiel, D. N. Neshev, W. Krolikowski, R. Fisher, A. Arie, and Y. S. Kivshar, Phys. Rev. Lett. 100, 103902(2008).
    [CrossRef]
  3. S. Saltiel, W. Krolikowski, D. N. Neshev, and Y. S. Kivshar, Opt. Express 15, 4132 (2007).
    [CrossRef]
  4. A. R. Tunyagi, M. Ulex, and K. Betzler, Phys. Rev. Lett. 90, 243901 (2003).
    [CrossRef]
  5. Y. Sheng, S. M. Saltiel, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, Opt. Lett. 35, 1317 (2010).
    [CrossRef]
  6. P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, Appl. Phys. Lett. 96, 261111 (2010).
    [CrossRef]
  7. K. Kalinowski, P. Roedig, Y. Sheng, M. Ayoub, J. Imbrock, C. Denz, and W. Krolikowski, Opt. Lett. 37, 1832 (2012).
    [CrossRef]
  8. P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Lett. 17, 447 (1970).
    [CrossRef]
  9. Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, Phys. Rev. Lett. 100, 163904 (2008).
    [CrossRef]
  10. A. Fragemann, V. Pasiskevicius, and F. Laurell, Appl. Phys. Lett. 85, 375 (2004).
    [CrossRef]
  11. Y. Sheng, A. Best, H. Butt, W. Krolikowski, A. Arie, and K. Koynov, Opt. Express 18, 16539 (2010).
    [CrossRef]
  12. X. Deng and X. Chen, Opt. Express 18, 15597 (2010).
    [CrossRef]
  13. K. Kalinowski, Q. Kong, V. Roppo, A. Arie, Y. Sheng, and W. Krolikowski, Appl. Phys. Lett. 99, 181128 (2011).
    [CrossRef]
  14. S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, IEEE J. Quantum Electron. 45, 1465 (2009).
    [CrossRef]
  15. X. Deng, H. Ren, Y. Zheng, and X. Chen, “Significantly enhanced second order nonlinearity in domain walls of ferroelectrics,” arXiv:1005.2925 (2010).
  16. N. Bloembergen, J. Opt. Soc. Am. 70, 1429 (1980).
    [CrossRef]
  17. R. W. Boyd, Nonlinear Optics (Academic, 2007).
  18. G. J. Edwards and M. Lawrence, Opt. Quantum Electron. 16, 373 (1984).
    [CrossRef]
  19. A. Zembrod, H. Puell, and J. A. Giordmaine, Optoelectron. Devices Technol. 1, 64 (1969).
    [CrossRef]
  20. V. Vacaitis, Opt. Commun. 209, 485 (2002).
    [CrossRef]

2012 (1)

2011 (1)

K. Kalinowski, Q. Kong, V. Roppo, A. Arie, Y. Sheng, and W. Krolikowski, Appl. Phys. Lett. 99, 181128 (2011).
[CrossRef]

2010 (4)

2009 (1)

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, IEEE J. Quantum Electron. 45, 1465 (2009).
[CrossRef]

2008 (2)

S. M. Saltiel, D. N. Neshev, W. Krolikowski, R. Fisher, A. Arie, and Y. S. Kivshar, Phys. Rev. Lett. 100, 103902(2008).
[CrossRef]

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, Phys. Rev. Lett. 100, 163904 (2008).
[CrossRef]

2007 (1)

2004 (1)

A. Fragemann, V. Pasiskevicius, and F. Laurell, Appl. Phys. Lett. 85, 375 (2004).
[CrossRef]

2003 (1)

A. R. Tunyagi, M. Ulex, and K. Betzler, Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef]

2002 (1)

V. Vacaitis, Opt. Commun. 209, 485 (2002).
[CrossRef]

1984 (1)

G. J. Edwards and M. Lawrence, Opt. Quantum Electron. 16, 373 (1984).
[CrossRef]

1980 (1)

1970 (1)

P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Lett. 17, 447 (1970).
[CrossRef]

1969 (2)

A. Zembrod, H. Puell, and J. A. Giordmaine, Optoelectron. Devices Technol. 1, 64 (1969).
[CrossRef]

E. Mathieu, ZAMP 20, 433 (1969).
[CrossRef]

Arie, A.

K. Kalinowski, Q. Kong, V. Roppo, A. Arie, Y. Sheng, and W. Krolikowski, Appl. Phys. Lett. 99, 181128 (2011).
[CrossRef]

Y. Sheng, S. M. Saltiel, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, Opt. Lett. 35, 1317 (2010).
[CrossRef]

Y. Sheng, A. Best, H. Butt, W. Krolikowski, A. Arie, and K. Koynov, Opt. Express 18, 16539 (2010).
[CrossRef]

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, IEEE J. Quantum Electron. 45, 1465 (2009).
[CrossRef]

S. M. Saltiel, D. N. Neshev, W. Krolikowski, R. Fisher, A. Arie, and Y. S. Kivshar, Phys. Rev. Lett. 100, 103902(2008).
[CrossRef]

Ayoub, M.

Bausa, L. E.

P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, Appl. Phys. Lett. 96, 261111 (2010).
[CrossRef]

Best, A.

Betzler, K.

A. R. Tunyagi, M. Ulex, and K. Betzler, Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef]

Bloembergen, N.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2007).

Butt, H.

Chen, X.

X. Deng and X. Chen, Opt. Express 18, 15597 (2010).
[CrossRef]

X. Deng, H. Ren, Y. Zheng, and X. Chen, “Significantly enhanced second order nonlinearity in domain walls of ferroelectrics,” arXiv:1005.2925 (2010).

Deng, X.

X. Deng and X. Chen, Opt. Express 18, 15597 (2010).
[CrossRef]

X. Deng, H. Ren, Y. Zheng, and X. Chen, “Significantly enhanced second order nonlinearity in domain walls of ferroelectrics,” arXiv:1005.2925 (2010).

Denz, C.

Edwards, G. J.

G. J. Edwards and M. Lawrence, Opt. Quantum Electron. 16, 373 (1984).
[CrossRef]

Fisher, R.

S. M. Saltiel, D. N. Neshev, W. Krolikowski, R. Fisher, A. Arie, and Y. S. Kivshar, Phys. Rev. Lett. 100, 103902(2008).
[CrossRef]

Fragemann, A.

A. Fragemann, V. Pasiskevicius, and F. Laurell, Appl. Phys. Lett. 85, 375 (2004).
[CrossRef]

Gao, Z. D.

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, Phys. Rev. Lett. 100, 163904 (2008).
[CrossRef]

Garcia, B. J.

P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, Appl. Phys. Lett. 96, 261111 (2010).
[CrossRef]

Giordmaine, J. A.

A. Zembrod, H. Puell, and J. A. Giordmaine, Optoelectron. Devices Technol. 1, 64 (1969).
[CrossRef]

Imbrock, J.

Kalinowski, K.

K. Kalinowski, P. Roedig, Y. Sheng, M. Ayoub, J. Imbrock, C. Denz, and W. Krolikowski, Opt. Lett. 37, 1832 (2012).
[CrossRef]

K. Kalinowski, Q. Kong, V. Roppo, A. Arie, Y. Sheng, and W. Krolikowski, Appl. Phys. Lett. 99, 181128 (2011).
[CrossRef]

Kivshar, Y.

Kivshar, Y. S.

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, IEEE J. Quantum Electron. 45, 1465 (2009).
[CrossRef]

S. M. Saltiel, D. N. Neshev, W. Krolikowski, R. Fisher, A. Arie, and Y. S. Kivshar, Phys. Rev. Lett. 100, 103902(2008).
[CrossRef]

S. Saltiel, W. Krolikowski, D. N. Neshev, and Y. S. Kivshar, Opt. Express 15, 4132 (2007).
[CrossRef]

Kong, Q.

K. Kalinowski, Q. Kong, V. Roppo, A. Arie, Y. Sheng, and W. Krolikowski, Appl. Phys. Lett. 99, 181128 (2011).
[CrossRef]

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, IEEE J. Quantum Electron. 45, 1465 (2009).
[CrossRef]

Krolikowski, W.

K. Kalinowski, P. Roedig, Y. Sheng, M. Ayoub, J. Imbrock, C. Denz, and W. Krolikowski, Opt. Lett. 37, 1832 (2012).
[CrossRef]

K. Kalinowski, Q. Kong, V. Roppo, A. Arie, Y. Sheng, and W. Krolikowski, Appl. Phys. Lett. 99, 181128 (2011).
[CrossRef]

Y. Sheng, A. Best, H. Butt, W. Krolikowski, A. Arie, and K. Koynov, Opt. Express 18, 16539 (2010).
[CrossRef]

Y. Sheng, S. M. Saltiel, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, Opt. Lett. 35, 1317 (2010).
[CrossRef]

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, IEEE J. Quantum Electron. 45, 1465 (2009).
[CrossRef]

S. M. Saltiel, D. N. Neshev, W. Krolikowski, R. Fisher, A. Arie, and Y. S. Kivshar, Phys. Rev. Lett. 100, 103902(2008).
[CrossRef]

S. Saltiel, W. Krolikowski, D. N. Neshev, and Y. S. Kivshar, Opt. Express 15, 4132 (2007).
[CrossRef]

Laurell, F.

A. Fragemann, V. Pasiskevicius, and F. Laurell, Appl. Phys. Lett. 85, 375 (2004).
[CrossRef]

Lawrence, M.

G. J. Edwards and M. Lawrence, Opt. Quantum Electron. 16, 373 (1984).
[CrossRef]

Martin, R. J.

P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Lett. 17, 447 (1970).
[CrossRef]

Mathieu, E.

E. Mathieu, ZAMP 20, 433 (1969).
[CrossRef]

Ming, N. B.

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, Phys. Rev. Lett. 100, 163904 (2008).
[CrossRef]

Molina, P.

P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, 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, IEEE J. Quantum Electron. 45, 1465 (2009).
[CrossRef]

S. M. Saltiel, D. N. Neshev, W. Krolikowski, R. Fisher, A. Arie, and Y. S. Kivshar, Phys. Rev. Lett. 100, 103902(2008).
[CrossRef]

S. Saltiel, W. Krolikowski, D. N. Neshev, and Y. S. Kivshar, Opt. Express 15, 4132 (2007).
[CrossRef]

Pasiskevicius, V.

A. Fragemann, V. Pasiskevicius, and F. Laurell, Appl. Phys. Lett. 85, 375 (2004).
[CrossRef]

Puell, H.

A. Zembrod, H. Puell, and J. A. Giordmaine, Optoelectron. Devices Technol. 1, 64 (1969).
[CrossRef]

Qi, Z.

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, Phys. Rev. Lett. 100, 163904 (2008).
[CrossRef]

Ramirez, M. O.

P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, Appl. Phys. Lett. 96, 261111 (2010).
[CrossRef]

Ren, H.

X. Deng, H. Ren, Y. Zheng, and X. Chen, “Significantly enhanced second order nonlinearity in domain walls of ferroelectrics,” arXiv:1005.2925 (2010).

Roedig, P.

Roppo, V.

K. Kalinowski, Q. Kong, V. Roppo, A. Arie, Y. Sheng, and W. Krolikowski, Appl. Phys. Lett. 99, 181128 (2011).
[CrossRef]

Saltiel, S.

Saltiel, S. M.

Y. Sheng, S. M. Saltiel, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, Opt. Lett. 35, 1317 (2010).
[CrossRef]

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, IEEE J. Quantum Electron. 45, 1465 (2009).
[CrossRef]

S. M. Saltiel, D. N. Neshev, W. Krolikowski, R. Fisher, A. Arie, and Y. S. Kivshar, Phys. Rev. Lett. 100, 103902(2008).
[CrossRef]

Sheng, Y.

K. Kalinowski, P. Roedig, Y. Sheng, M. Ayoub, J. Imbrock, C. Denz, and W. Krolikowski, Opt. Lett. 37, 1832 (2012).
[CrossRef]

K. Kalinowski, Q. Kong, V. Roppo, A. Arie, Y. Sheng, and W. Krolikowski, Appl. Phys. Lett. 99, 181128 (2011).
[CrossRef]

Y. Sheng, A. Best, H. Butt, W. Krolikowski, A. Arie, and K. Koynov, Opt. Express 18, 16539 (2010).
[CrossRef]

Y. Sheng, S. M. Saltiel, W. Krolikowski, A. Arie, K. Koynov, and Y. Kivshar, Opt. Lett. 35, 1317 (2010).
[CrossRef]

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, IEEE J. Quantum Electron. 45, 1465 (2009).
[CrossRef]

Tien, P. K.

P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Lett. 17, 447 (1970).
[CrossRef]

Tunyagi, A. R.

A. R. Tunyagi, M. Ulex, and K. Betzler, Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef]

Ulex, M.

A. R. Tunyagi, M. Ulex, and K. Betzler, Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef]

Ulrich, R.

P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Lett. 17, 447 (1970).
[CrossRef]

Vacaitis, V.

V. Vacaitis, Opt. Commun. 209, 485 (2002).
[CrossRef]

Voloch-Bloch, N.

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, IEEE J. Quantum Electron. 45, 1465 (2009).
[CrossRef]

Zembrod, A.

A. Zembrod, H. Puell, and J. A. Giordmaine, Optoelectron. Devices Technol. 1, 64 (1969).
[CrossRef]

Zhang, Y.

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, Phys. Rev. Lett. 100, 163904 (2008).
[CrossRef]

Zheng, Y.

X. Deng, H. Ren, Y. Zheng, and X. Chen, “Significantly enhanced second order nonlinearity in domain walls of ferroelectrics,” arXiv:1005.2925 (2010).

Zhu, S. N.

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, Phys. Rev. Lett. 100, 163904 (2008).
[CrossRef]

Appl. Phys. Lett. (4)

P. Molina, M. O. Ramirez, B. J. Garcia, and L. E. Bausa, Appl. Phys. Lett. 96, 261111 (2010).
[CrossRef]

P. K. Tien, R. Ulrich, and R. J. Martin, Appl. Phys. Lett. 17, 447 (1970).
[CrossRef]

A. Fragemann, V. Pasiskevicius, and F. Laurell, Appl. Phys. Lett. 85, 375 (2004).
[CrossRef]

K. Kalinowski, Q. Kong, V. Roppo, A. Arie, Y. Sheng, and W. Krolikowski, Appl. Phys. Lett. 99, 181128 (2011).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. M. Saltiel, Y. Sheng, N. Voloch-Bloch, D. N. Neshev, W. Krolikowski, A. Arie, K. Koyonov, and Y. S. Kivshar, IEEE J. Quantum Electron. 45, 1465 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

V. Vacaitis, Opt. Commun. 209, 485 (2002).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

G. J. Edwards and M. Lawrence, Opt. Quantum Electron. 16, 373 (1984).
[CrossRef]

Optoelectron. Devices Technol. (1)

A. Zembrod, H. Puell, and J. A. Giordmaine, Optoelectron. Devices Technol. 1, 64 (1969).
[CrossRef]

Phys. Rev. Lett. (3)

Y. Zhang, Z. D. Gao, Z. Qi, S. N. Zhu, and N. B. Ming, Phys. Rev. Lett. 100, 163904 (2008).
[CrossRef]

A. R. Tunyagi, M. Ulex, and K. Betzler, Phys. Rev. Lett. 90, 243901 (2003).
[CrossRef]

S. M. Saltiel, D. N. Neshev, W. Krolikowski, R. Fisher, A. Arie, and Y. S. Kivshar, Phys. Rev. Lett. 100, 103902(2008).
[CrossRef]

ZAMP (1)

E. Mathieu, ZAMP 20, 433 (1969).
[CrossRef]

Other (2)

R. W. Boyd, Nonlinear Optics (Academic, 2007).

X. Deng, H. Ren, Y. Zheng, and X. Chen, “Significantly enhanced second order nonlinearity in domain walls of ferroelectrics,” arXiv:1005.2925 (2010).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Phase-matching diagram of Čerenkov second-harmonic generation. (b) Schematic of the simulation with SHG in optical media containing two layers of different nonlinear responses: χ(2) and χ˜(2).

Fig. 2.
Fig. 2.

Near- (left column) and far-field (right column) intensity of the SHG in composite media with fundamental beam propagating along boundary separating different strengths of the nonlinearity.

Fig. 3.
Fig. 3.

(a) Illustrating different spatial profiles of the χ(2) modulation across the boundary. (b) The angular profile of the SH emission. Two peaks located at θ=±28.6° represent Čerenkov signals. Notice the strength of the emission increasing with D.

Fig. 4.
Fig. 4.

(a) The SHG as a function of incident angle α of the fundamental wave. The Čerenkov emission is represented by the two outmost curves. The central line depicts the forward SH signal. (b) The effect of the beam width w on the Čerenkov SHG. The calculations correspond to D=109.

Equations (2)

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

E1z=i2k12E1iω12χ(2)(x)k1c2E1*E2ei(k22k1)z,E2z=i2k22E2iω22χ(2)(x)2k2c2E12ei(2k1k2)z,
E=E1(x,z)ei(k1zω1t)+E2(x,z)ei(k2zω2t)+c.c.

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