Abstract

The evolution of light pulses and beams in a quasi-phase-matched (QPM) quadratic medium is usually described by considering only the spatial harmonic of the QPM grating that minimizes the residual phase-mismatch. I show that, for strongly phase-mismatched interactions (the cascading regime), several harmonics need to be accounted for in order to obtain the correct value of the effective cubic nonlinearity, for which I find a simple analytical expression. I discuss the effects of the higher order harmonics of the grating on solitary wave propagation.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, Opt. Lett. 17, 28 (1992).
    [CrossRef]
  2. G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
    [CrossRef]
  3. P. Baldi, L. Chanvillard, P. Aschieri, J. Pruvot, M. P. De Micheli, D. B. Ostrowsky, and G. Bellanca, Electron. Lett. 35, 217 (1999).
    [CrossRef]
  4. R. Schiek, L. Friedrich, H. Fang, G. I. Stegeman, K. R. Parameswaran, M.-H. Chou, and M. M. Fejer, Opt. Lett. 24, 1617 (1999).
    [CrossRef]
  5. B. Bourliaguet, V. Couderc, A. Barthelemy, G. W. Ross, P. G. R. Smith, D. C. Hanna, and C. De Angelis, Opt. Lett. 24, 1410 (1999).
    [CrossRef]
  6. C. R. Menyuk, R. Schiek, and L. Torner, J. Opt. Soc. Am. B 11, 2434 (1994).
    [CrossRef]
  7. M. Bache, J. Moses, and F. W. Wise, J. Opt. Soc. Am. B 24, 2752 (2007).
    [CrossRef]
  8. B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, Phys. Rev. Lett. 109, 04392 (2012).
    [CrossRef]
  9. M. Levenius, M. Conforti, F. Baronio, V. Pasiskevicius, F. Laurell, C. De Angelis, and K. Gallo, Opt. Lett. 37, 1727 (2012).
    [CrossRef]
  10. U. Sapaev and D. T. Reid, Opt. Express 13, 3264 (2005).
    [CrossRef]
  11. F. Baronio, C. De Angelis, M. Marangoni, C. Manzoni, R. Ramponi, and G. Cerullo, Opt. Express 14, 4774 (2006).
    [CrossRef]
  12. L. Kornaszewski, K. Kohler, U. Sapaev, and D. T. Reid, Opt. Lett. 33, 378 (2008).
    [CrossRef]
  13. M. Conforti, F. Baronio, and C. De Angelis, Opt. Lett. 32, 1779 (2007).
    [CrossRef]
  14. M. Marangoni, D. Brida, M. Conforti, A. D. Capobianco, C. Manzoni, F. Baronio, G. F. Nalesso, C. De Angelis, R. Ramponi, and G. Cerullo, Opt. Lett. 34, 241 (2009).
    [CrossRef]
  15. C. R. Phillips, L. G. Gallman, and M. M. Fejer, Opt. Express 21, 10139 (2013).
    [CrossRef]
  16. C. Langrock, M. M. Fejer, and M. Fermann, Opt. Lett. 32, 2478 (2007).
    [CrossRef]
  17. C. R. Phillips, C. Langrock, J. S. Pelec, M. M. Fejer, I. Hartl, and M. E. Fermann, Opt. Express 19, 18754 (2011).
    [CrossRef]
  18. M. Conforti, F. Baronio, and C. De Angelis, IEEE Photon. J. 2, 600 (2010).
    [CrossRef]
  19. C. B. Clausen, O. Bang, and Y. Kivshar, Phys. Rev. Lett 78, 4749 (1997).
    [CrossRef]
  20. A. Kobyakov, F. Lederer, O. Bang, and Y. S. Kivshar, Opt. Lett. 23, 506 (1998).
    [CrossRef]
  21. O. Bang, C. B. Clausen, P. I. Christiansen, and L. Torner, Opt. Lett. 24, 1413 (1999).
    [CrossRef]
  22. D. H. Jundt, Opt. Lett. 22, 1553 (1997).
    [CrossRef]

2013

2012

2011

2010

M. Conforti, F. Baronio, and C. De Angelis, IEEE Photon. J. 2, 600 (2010).
[CrossRef]

2009

2008

2007

2006

2005

1999

1998

1997

D. H. Jundt, Opt. Lett. 22, 1553 (1997).
[CrossRef]

C. B. Clausen, O. Bang, and Y. Kivshar, Phys. Rev. Lett 78, 4749 (1997).
[CrossRef]

1996

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

1994

1992

Aschieri, P.

P. Baldi, L. Chanvillard, P. Aschieri, J. Pruvot, M. P. De Micheli, D. B. Ostrowsky, and G. Bellanca, Electron. Lett. 35, 217 (1999).
[CrossRef]

Bache, M.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, Phys. Rev. Lett. 109, 04392 (2012).
[CrossRef]

M. Bache, J. Moses, and F. W. Wise, J. Opt. Soc. Am. B 24, 2752 (2007).
[CrossRef]

Baldi, P.

P. Baldi, L. Chanvillard, P. Aschieri, J. Pruvot, M. P. De Micheli, D. B. Ostrowsky, and G. Bellanca, Electron. Lett. 35, 217 (1999).
[CrossRef]

Bang, O.

Baronio, F.

Barthelemy, A.

Bellanca, G.

P. Baldi, L. Chanvillard, P. Aschieri, J. Pruvot, M. P. De Micheli, D. B. Ostrowsky, and G. Bellanca, Electron. Lett. 35, 217 (1999).
[CrossRef]

Bourliaguet, B.

Brida, D.

Capobianco, A. D.

Cerullo, G.

Chanvillard, L.

P. Baldi, L. Chanvillard, P. Aschieri, J. Pruvot, M. P. De Micheli, D. B. Ostrowsky, and G. Bellanca, Electron. Lett. 35, 217 (1999).
[CrossRef]

Chong, A.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, Phys. Rev. Lett. 109, 04392 (2012).
[CrossRef]

Chou, M.-H.

Christiansen, P. I.

Clausen, C. B.

O. Bang, C. B. Clausen, P. I. Christiansen, and L. Torner, Opt. Lett. 24, 1413 (1999).
[CrossRef]

C. B. Clausen, O. Bang, and Y. Kivshar, Phys. Rev. Lett 78, 4749 (1997).
[CrossRef]

Conforti, M.

Couderc, V.

De Angelis, C.

De Micheli, M. P.

P. Baldi, L. Chanvillard, P. Aschieri, J. Pruvot, M. P. De Micheli, D. B. Ostrowsky, and G. Bellanca, Electron. Lett. 35, 217 (1999).
[CrossRef]

DeSalvo, R.

Fang, H.

Fejer, M. M.

Fermann, M.

Fermann, M. E.

Friedrich, L.

Gallman, L. G.

Gallo, K.

Hagan, D. J.

Hanna, D. C.

Hartl, I.

Jundt, D. H.

Kivshar, Y.

C. B. Clausen, O. Bang, and Y. Kivshar, Phys. Rev. Lett 78, 4749 (1997).
[CrossRef]

Kivshar, Y. S.

Kobyakov, A.

Kohler, K.

Kornaszewski, L.

Langrock, C.

Laurell, F.

Lederer, F.

Levenius, M.

Manzoni, C.

Marangoni, M.

Menyuk, C. R.

Moses, J.

Nalesso, G. F.

Ostrowsky, D. B.

P. Baldi, L. Chanvillard, P. Aschieri, J. Pruvot, M. P. De Micheli, D. B. Ostrowsky, and G. Bellanca, Electron. Lett. 35, 217 (1999).
[CrossRef]

Parameswaran, K. R.

Pasiskevicius, V.

Pelec, J. S.

Phillips, C. R.

Pruvot, J.

P. Baldi, L. Chanvillard, P. Aschieri, J. Pruvot, M. P. De Micheli, D. B. Ostrowsky, and G. Bellanca, Electron. Lett. 35, 217 (1999).
[CrossRef]

Ramponi, R.

Reid, D. T.

Ross, G. W.

Sapaev, U.

Schiek, R.

Sheik-Bahae, M.

Smith, P. G. R.

Stegeman, G.

Stegeman, G. I.

Torner, L.

Van Stryland, E. W.

Vanherzeele, H.

Wise, F. W.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, Phys. Rev. Lett. 109, 04392 (2012).
[CrossRef]

M. Bache, J. Moses, and F. W. Wise, J. Opt. Soc. Am. B 24, 2752 (2007).
[CrossRef]

Zhou, B. B.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, Phys. Rev. Lett. 109, 04392 (2012).
[CrossRef]

Electron. Lett.

P. Baldi, L. Chanvillard, P. Aschieri, J. Pruvot, M. P. De Micheli, D. B. Ostrowsky, and G. Bellanca, Electron. Lett. 35, 217 (1999).
[CrossRef]

IEEE Photon. J.

M. Conforti, F. Baronio, and C. De Angelis, IEEE Photon. J. 2, 600 (2010).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

G. I. Stegeman, D. J. Hagan, and L. Torner, Opt. Quantum Electron. 28, 1691 (1996).
[CrossRef]

Phys. Rev. Lett

C. B. Clausen, O. Bang, and Y. Kivshar, Phys. Rev. Lett 78, 4749 (1997).
[CrossRef]

Phys. Rev. Lett.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, Phys. Rev. Lett. 109, 04392 (2012).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

Effective cascading nonlinearity γ normalized to the homogeneous medium nonlinearity γ0 (solid green line). Solid blue curve, exact value Eq. (7); dashed red curve, standard first-order QPM approximation [Eq. (10), m=1]; dashed–dotted black curve, third-order QPM approximation [Eq. (10), m=3].

Fig. 2.
Fig. 2.

Relative percentage error between exact cascading nonlinearity [Eq. (7)] and first-order QPM approximation [Eq. (10), m=1]. Dotted red curves show contour plots of residual phase-mismatch δk for three values (50,0,70)mm1. Dispersion from Sellmeier equation [22] and T=100°C.

Fig. 3.
Fig. 3.

Soliton peak intensity for a duration T0=40fs (λ0=1500nm) as a function of QPM period. Solid blue curve, exact value; Dashed red curve, first-order QPM approximation; Dashed–dotted black curve, third-order QPM approximation.

Fig. 4.
Fig. 4.

Relative error for the soliton peak intensity: solid blue curve, single-harmonic first-order QPM approximation; dashed red curve, two-harmonic first-order QPM approximation.

Fig. 5.
Fig. 5.

(a) Input (dashed curves) and output (solid curves) intensities from numerical solutions of Eqs. (1) and (2) for a sech pulse of duration T0=40fs and wavelength λ0=1500nm, after propagation in a 4-cm-long crystal (around 3 dispersion lengths), poled with period Λ=70μm. Thick curves, soliton case; thin curves, quasi-linear regime (half input intensity). Black dots show the output of the broadband χ(2) model [18]. (b) Exact cascading (blue curve) and material Kerr (red curve near zero) nonlinearities as a function of wavelength. The spectral shape of the soliton, described in (a), is superimposed for reference (dashed curve).

Equations (10)

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

iA1zβ122A1t2+g(z)χA1*A2eiΔkz=0,
iA2z+iδA2tβ222A2t2+g(z)χA12eiΔkz=0,
g(z)==+2iπ(2+1)ei(2+1)κz,κ=2πΛ
A22χiπA12m=oddei(κmΔk)z1m(κmΔk).
NL=(2χπ)2|A1|2A1(m,n)=oddei(m+n)κzei(mκ+Δk)zmn(nκΔk).
iA1zβ122A1t2+γ|A1|2A1=0,
γ=(2χπ)2n=odd1n2(Δknκ)=(2χπ)2S.
S={π24Δk,κ=0π24Δkπ2κΔk2tan(π2Δkκ),κ0.
γ0=χ2Δk,
γm=(2χmπ)21δk,δk=Δkm2πΛ.

Metrics