## Complete spatial and temporal locking in phase-mismatched second-harmonic generation

Optics Express, Vol. 17, Issue 5, pp. 3141-3147 (2009)

http://dx.doi.org/10.1364/OE.17.003141

Acrobat PDF (520 KB)

### Abstract

We experimentally demonstrate simultaneous phase and group velocity locking of fundamental and generated second harmonic pulses in Lithium Niobate, under conditions of material phase mismatch. In phase-mismatched, pulsed second harmonic generation in addition to a reflected signal two forward-propagating pulses are also generated at the interface between a linear and a second order nonlinear material: the first pulse results from the solution of the homogeneous wave equation, and propagates at the group velocity expected from material dispersion; the second pulse is the solution of the inhomogeneous wave equation, is phase-locked and trapped by the pump pulse, and follows the pump trajectory. At normal incidence, the normal and phase locked pulses simply trail each other. At oblique incidence, the consequences can be quite dramatic. The homogeneous pulse refracts as predicted by material dispersion and Snell’s law, yielding at least two spatially separate second harmonic spots at the medium’s exit. We thus report the first experimental results showing that, at oblique incidence, fundamental and phase-locked second harmonic pulses travel with the same group velocity and follow the same trajectory. This is direct evidence that, at least up to first order, the effective dispersion of the phase-locked pulse is similar to the dispersion of the pump pulse.

© 2009 Optical Society of America

## 1. Introduction

5. A. Feisst and P. Koidl, “Current induced periodic ferroelectric domain structures in LiNbO_{3} applied for efficient nonlinear optical frequency mixing”, Appl. Phys. Lett. **47**, 1125 (1985). [CrossRef]

8. S. K. Kurtz and T. T. Perry, “A Powder Technique for the Evaluation of Nonlinear Optical Materials”, J. Appl. Phys. **39**, 3798 (1968). [CrossRef]

9. J. P. van der Ziel, “Phase-matched harmonic generation in a laminar structure with wave propagation in the plane of the layers”, Appl. Phys. Lett. **26**, 60 (1976). [CrossRef]

11. R. J. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP”, Opt. Lett. **17**, 28 (1992). [CrossRef] [PubMed]

12. N. C. Kothari and X. Carlotti, “Transient second-harmonic generation: influence of effective group-velocity dispersion”, J. Opt. Soc. Am. B **5**, 756 (1988). [CrossRef]

16. D. Noordam, H.J. Bakker, M. P. de Boer, and H.B. van Linden van den Heuvell, “Second-harmonic generation of femtosecond pulses: observation of phase-mismatch effects”, Opt. Lett. **15**, 1464 (1990)
[CrossRef] [PubMed]

17. M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, “Second Harmonic Generation of Femtosecond Pulses at the Boundary of a Nonlinear Dielectric”, Phys. Rev. Lett. **83**, 2934 (1999). [CrossRef]

18. W. Su, L. Qian, H. Luo, X. Fu, H. Zhu, T. Wang, K. Beckwitt, Y. Chen, and F. Wise, “Induced group-velocity dispersion in phase-mismatched second-harmonic generation”, J. Opt. Soc. Am. B **23**, 51 (2006). [CrossRef]

19. V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J.W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media”, Phys. Rev. A **76**, 033829 (2007)
[CrossRef]

20. M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J.W. Haus, J.V. Foreman, N. Akozbek, M.J. Bloemer, and M. Scalora, “Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation”, Phys. Rev. Lett. **101**, 113905 (2008)
[CrossRef] [PubMed]

20. M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J.W. Haus, J.V. Foreman, N. Akozbek, M.J. Bloemer, and M. Scalora, “Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation”, Phys. Rev. Lett. **101**, 113905 (2008)
[CrossRef] [PubMed]

19. V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J.W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media”, Phys. Rev. A **76**, 033829 (2007)
[CrossRef]

_{eff}=c<k>/<ω> as the ratio of expectation values

19. V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J.W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media”, Phys. Rev. A **76**, 033829 (2007)
[CrossRef]

_{eff}is mapped onto the pump index of refraction, the imaginary part of the index is obtained by performing a Kramers-Kronig reconstruction, which in turn yields an imaginary part for the second harmonic field that is identical to the index of refraction for the pump field. Therefore, it is sufficient to tune the pump to a region of transparency to guarantee suppression of absorption at the harmonic wavelengths.

## 2. Numerical simulations

20. M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J.W. Haus, J.V. Foreman, N. Akozbek, M.J. Bloemer, and M. Scalora, “Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation”, Phys. Rev. Lett. **101**, 113905 (2008)
[CrossRef] [PubMed]

*z*,

*y*,

*t*), 𝓗

^{ℓω}(

*z*,

*y*,

*t*) are generic, spatially- and temporally-dependent, complex envelope functions;

*k*and

*ω*are carrier wave vector and frequency, respectively, and ℓ is an integer. Eqs.(1) are a convenient representation of the fields, and no

*a priori*assumptions are made about the field envelopes. We have also assumed that a TM-polarized incident field generates similarly polarized harmonics. The linear response of the medium is described by a Lorentz oscillator model:

*μ*(

*ω*) = 1, where

*γ*,

*ω*, and

_{p}*ω*are the damping coefficient, the plasma and resonance frequencies, respectively. The second order nonlinear polarization is:

_{r}*P*=

_{NL}*χ*

^{(2)}

*E*

^{2}. Expanding the field into its components yields nonlinear polarization terms at the fundamental and second harmonic frequencies: 𝒫 (

*z*,

*t*)= 2

*χ*

^{(2)}

_{ω}ℰ

^{*}

_{ω}and ℰ

_{2ω}(

*z*,

*t*) =

*χ*

^{(2)}

_{2ω}ℰ

^{2}

_{ω}. Assuming that polarization and currents may be decomposed as in Eqs.(1), we obtain the following Maxwell-Lorentz system of equations for the ℓ

^{th}field components, in the scaled two-dimensional space (

*y*̃,

*ξ*) plus time (

*τ*) coordinate system:

_{NL}refer to linear electric currents, polarization, and nonlinear polarization, respectively. The coordinates are scaled so that

*ξ*=

*z*/

*λ*

_{0},

*y*̃=

*y*/

*λ*

_{0},

*τ*=

*ct*/

*λ*

_{0},

*λ*

_{0}=1

*μm*is just reference wavelength;

*γ*,

*β*= 2

_{ℓω}*π*ℓ

*ω*/

*ω*

_{0},

*β*= 2

*π*

*ω*/

_{r}*ω*

_{0},

*ω*, are the scaled damping coefficient, wave-vector, resonance and electric plasma frequencies for the ℓ

_{p}^{th}harmonic, respectively.

*θ*is the angle of incidence of the pump field. The equations are solved using a split-step, fast Fourier transform-based pulse propagation algorithm [20

_{i}**101**, 113905 (2008)
[CrossRef] [PubMed]

2. N. Bloembergen and P. S. Pershan, “Light Waves at the Boundary of Nonlinear Media”, Phys. Rev. **128**, 606 (1962). [CrossRef]

21. P. D. Marker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of Dispersion and Focusing on the Production of Optical Harmonics”, Phys. Rev. Lett. **8**, 21 (1962). [CrossRef]

## 3. Experiment

^{2}. A special prismatic shape of the sample ensured the output face to be 20° tilted from the propagation direction. Thus, this sample acts simultaneously as a generator and a spectrum analyser, as each wavelength refracts at different angle and according to Snell’s law.

## 4. Phase-velocity locking

_{ee-e}=2.45·10

^{4}cm

^{-1}). One of the beams (B) exits at the angle that corresponds to n

_{e}(400nm), while the other (C) exits exactly overlapped with the pump beam, i.e. at the angle that corresponds to n

_{e}(800nm). This result demonstrates that the generated second harmonic signal separates in two different parts, one freely propagating without further interaction with the pump pulse, the other locked with the pump, propagating exactly at the same phase-velocity (A-C), i.e. having exactly the same refractive index and following exactly the same trajectory as the pump. It should be mentioned here that an unexpected high conversion efficiency was observed for the locked second-harmonic beam, whose average power was of the order of 10

^{-3}-10

^{-2}of the pump one.

_{oo-e}=1.20·10

^{4}cm

^{-1}). The generated SH signal is once again broken in two fragments, one freely propagating (E), and the other (F) once again phase-locked to the pump (D-F) that exits the lithium niobate crystal at the same angle as the fundamental frequency. Also in this case there is a transfer of refractive index from the generating to the generated wavelength: in fact the e-polarised locked pulse at 400nm experiences exactly the ordinary refractive index of 800nm pump pulses.

_{oe-e}=1.8·10

^{4}cm

^{-1}, oe-o, Δk

_{oe-o}=3.5·10

^{4}cm

^{-1}) couplings are obtained simultaneously. The fundamental e-polarisation (G) generates two e-polarised second harmonic pulses (M, O) according to the type-0 coupling. The fundamental o-polarisation generates two e-polarised second harmonic pulses according to type-I coupling, (M, N). The temporal overlapping of the fundamental o- and e-polarisations generates a second harmonic signal (L) by type-II coupling, which is the only second harmonic o-polarisation. Indeed, due to group velocity mismatch of the two pumps, the phase-locked component cannot be sustained because the two cross-polarised pump pulses rapidly separate and the inhomogeneous (forced) term in the second harmonic field equation vanishes. Thus, only the pulse associated with the homogenous term is recorded at the output.

## 5. Group-velocity locking

*ω+2ω=3ω*. This interaction is possible only if the ω and 2ω pulses arrive simultaneously onto the BBO crystal, and its detection would certainly demonstrate that group-velocity locking was taking place inside the LNB crystal. Indeed, using our setup a generated ultraviolet (3ω) pulse train was monitored by the PMT on the scope screen (Figure 4(I)).

## 6. Conclusions

## Acknowledgments

*La Sapienza*University with the project “professori visitatori”. A.M.D.G.

## References and links

1. | J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric”, Phys. Rev. |

2. | N. Bloembergen and P. S. Pershan, “Light Waves at the Boundary of Nonlinear Media”, Phys. Rev. |

3. | P. D. Maker, R. W. Terhune, N. Nisenoff, and C. M. Savage, “Effects of Dispersion and Focusing on the Production of Optical Harmonics”, Phys. Rev. Lett. |

4. | N. Bloembergen, H. J. Simon, and C. H. Lee, “Total Reflection Phenomena in Second-Harmonic Generation of Light”, Phys. Rev. |

5. | A. Feisst and P. Koidl, “Current induced periodic ferroelectric domain structures in LiNbO |

6. | U. Osterberg and W. Margulis, “Dye laser pumped by Nd: YAG laser pulses frequency doubled in a glass optical fiber”, Opt. Lett. |

7. | G. A. Magel, M. M. Fejer, and R. L. Byer, “Quasi-phase-matched second-harmonic generation of blue light in periodically poled LiNbO |

8. | S. K. Kurtz and T. T. Perry, “A Powder Technique for the Evaluation of Nonlinear Optical Materials”, J. Appl. Phys. |

9. | J. P. van der Ziel, “Phase-matched harmonic generation in a laminar structure with wave propagation in the plane of the layers”, Appl. Phys. Lett. |

10. | K. Sakoda and K. Othaka, “Sum-frequency generation in a two-dimensional photonic lattice”, Phys. Rev. B |

11. | R. J. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP”, Opt. Lett. |

12. | N. C. Kothari and X. Carlotti, “Transient second-harmonic generation: influence of effective group-velocity dispersion”, J. Opt. Soc. Am. B |

13. | R. Maleck Rassoul, A. Ivanov, E. Freysz, A. Ducasse, and F. Hache, “Second-harmonic generation under phase-velocity and group-velocity mismatch:influence of cascading self-phase and cross-phase modulation”, Opt. Lett. |

14. | S. Cussat-Blanc, R. Maleck-Rassoul, A. Ivanov, E. Freysz, and A. Ducasse, “Influence of cascading phenomena on a type I second-harmonic wave generated by an intense femtosecond pulse: application to the measurement of the effective second-order coefficient”, Opt. Lett. |

15. | E. Fazio, M. Zitelli, S. Dominici, C. Sibilia, G. D’Aguanno, and M. Bertolotti, “Phase-driven pulse
”, Opt. Commun. |

16. | D. Noordam, H.J. Bakker, M. P. de Boer, and H.B. van Linden van den Heuvell, “Second-harmonic generation of femtosecond pulses: observation of phase-mismatch effects”, Opt. Lett. |

17. | M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, “Second Harmonic Generation of Femtosecond Pulses at the Boundary of a Nonlinear Dielectric”, Phys. Rev. Lett. |

18. | W. Su, L. Qian, H. Luo, X. Fu, H. Zhu, T. Wang, K. Beckwitt, Y. Chen, and F. Wise, “Induced group-velocity dispersion in phase-mismatched second-harmonic generation”, J. Opt. Soc. Am. B |

19. | V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J.W. Haus, O. G. Kosareva, and V. P. Kandidov, “Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media”, Phys. Rev. A |

20. | M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J.W. Haus, J.V. Foreman, N. Akozbek, M.J. Bloemer, and M. Scalora, “Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation”, Phys. Rev. Lett. |

21. | P. D. Marker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of Dispersion and Focusing on the Production of Optical Harmonics”, Phys. Rev. Lett. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4350) Nonlinear optics : Nonlinear optics at surfaces

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 3, 2008

Revised Manuscript: December 11, 2008

Manuscript Accepted: January 20, 2009

Published: February 17, 2009

**Citation**

Eugenio Fazio, Federico Pettazzi, Marco Centini, Mathieu Chauvet, Alessandro Belardini, Massimo Alonzo, Concita Sibilia, Mario Bertolotti, and Micheal Scalora, "Complete spatial and temporal locking in phase-mismatched
second-harmonic generation," Opt. Express **17**, 3141-3147 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3141

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### References

- J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127, 1918 (1962). [CrossRef]
- N. Bloembergen and P. S. Pershan, "Light Waves at the Boundary of Nonlinear Media," Phys. Rev. 128, 606 (1962). [CrossRef]
- P. D. Maker, R. W. Terhune, N. Nisenoff, and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics," Phys. Rev. Lett. 8, 21 (1962). [CrossRef]
- N. Bloembergen, H. J. Simon, and C. H. Lee, "Total Reflection Phenomena in Second-Harmonic Generation of Light," Phys. Rev. 181, 1261 (1969). [CrossRef]
- A. Feisst and P. Koidl, "Current induced periodic ferroelectric domain structures in LiNbO3 applied for efficient nonlinear optical frequency mixing," Appl. Phys. Lett. 47, 1125 (1985). [CrossRef]
- U. Osterberg and W. Margulis, "Dye laser pumped by Nd: YAG laser pulses frequency doubled in a glass optical fiber," Opt. Lett. 11, 516 (1986). [CrossRef] [PubMed]
- G. A. Magel, M. M. Fejer, and R. L. Byer, "Quasi-phase-matched second-harmonic generation of blue light in periodically poled LiNbO3," Appl. Phys. Lett. 56, 108 (1990). [CrossRef]
- S. K. Kurtz and T. T. Perry, "A Powder Technique for the Evaluation of Nonlinear Optical Materials," J. Appl. Phys. 39, 3798 (1968). [CrossRef]
- J. P. van der Ziel, "Phase−matched harmonic generation in a laminar structure with wave propagation in the plane of the layers," Appl. Phys. Lett. 26, 60 (1976). [CrossRef]
- K. Sakoda and K. Othaka, "Sum-frequency generation in a two-dimensional photonic lattice," Phys. Rev. B 54, 5742 (1996). [CrossRef]
- R. J. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, E. W. Van Stryland, and H. Vanherzeele, "Self-focusing and self-defocusing by cascaded second-order effects in KTP," Opt. Lett. 17, 28 (1992). [CrossRef] [PubMed]
- N. C. Kothari and X. Carlotti, "Transient second-harmonic generation: influence of effective group-velocity dispersion," J. Opt. Soc. Am. B 5, 756 (1988). [CrossRef]
- R. Maleck Rassoul, A. Ivanov, E. Freysz, A. Ducasse, and F. Hache, "Second-harmonic generation under phase-velocity and group-velocity mismatch:influence of cascading self-phase and cross-phase modulation," Opt. Lett. 22, 268 (1997). [CrossRef] [PubMed]
- S. Cussat-Blanc, R. Maleck-Rassoul, A. Ivanov, E. Freysz, and A. Ducasse, "Influence of cascading phenomena on a type I second-harmonic wave generated by an intense femtosecond pulse: application to the measurement of the effective second-order coefficient," Opt. Lett. 23, 1585 (1998). [CrossRef]
- E. Fazio, M. Zitelli, S. Dominici, C. Sibilia, G. D'Aguanno, and M. Bertolotti, "Phase-driven pulse breaking during perfectly-matched second harmonic generation," Opt. Commun. 148, 427 (1998). [CrossRef]
- D. Noordam, H. J. Bakker, M. P. de Boer, H. B. van Linden van den Heuvell, "Second-harmonic generation of femtosecond pulses: observation of phase-mismatch effects," Opt. Lett. 15, 1464 (1990) [CrossRef] [PubMed]
- M. Mlejnek, E. M. Wright, J. V. Moloney, and N. Bloembergen, "Second Harmonic Generation of Femtosecond Pulses at the Boundary of a Nonlinear Dielectric", Phys. Rev. Lett. 83, 2934 (1999). [CrossRef]
- W. Su, L. Qian, H. Luo, X. Fu, H. Zhu, T. Wang, K. Beckwitt, Y. Chen, and F. Wise, "Induced group-velocity dispersion in phase-mismatched second-harmonic generation", J. Opt. Soc. Am. B 23, 51 (2006). [CrossRef]
- V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. J. Bloemer, J. W. Haus, O. G. Kosareva, V. P. Kandidov, "Role of phase matching in pulsed second-harmonic generation: Walk-off and phase-locked twin pulses in negative-index media", Phys. Rev. A 76, 033829 (2007) [CrossRef]
- M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer, M. Scalora, "Inhibition of Linear Absorption in Opaque Materials Using Phase-Locked Harmonic Generation", Phys. Rev. Lett. 101, 113905 (2008) [CrossRef] [PubMed]
- P. D. Marker, R. W. Terhune, M. Nisenoff and C. M. Savage, "Effects of Dispersion and Focusing on the Production of Optical Harmonics", Phys. Rev. Lett. 8, 21 (1962). [CrossRef]

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