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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 6 — Mar. 14, 2011
  • pp: 5213–5218
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Dependences of the group velocity for femtosecond pulses in MgO-doped PPLN crystal

Yu-Ping Chen, Wen-Jie Lu, Yu-Xing Xia, and Xian-Feng Chen  »View Author Affiliations


Optics Express, Vol. 19, Issue 6, pp. 5213-5218 (2011)
http://dx.doi.org/10.1364/OE.19.005213


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Abstract

Theoretical investigation on the group velocity control of ultrafast pulses through quadratic cascading nonlinear interaction is presented. The dependences of the fractional time delay as well as the quality factor of the delayed femtosecond pulse on the peak intensity, group velocity mismatch, wave-vector mismatch and the pulse duration are examined. The results may help to understand to what extent some optical operation parameters could have played a role in controlling the ultrashort pulses. We also predict the maximum achievable pulse delay or advancement efficiency without large distortions. A compact solid medium integrating multiple functions including slowing light, wavelength conversion or broadcasting on a single chip, may bring significant practicality and high integration applications at optical communication band.

© 2011 OSA

1. Introduction

Many interests have been focused on manipulating the group velocity of light pulses [1

1. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]

4

4. Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, “Large tunable optical delays via self-phase modulation and dispersion,” Opt. Express 14(25), 12022–12027 (2006). [CrossRef] [PubMed]

]. Its main goals include the understanding of physical laws for light pulse propagation as well as many promising applications such as controllable optical delay lines, all-optical memories, and data resynchronization devices for all optical communication [5

5. R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Applications of Slow Light in Telecommunications,” Opt. Photon. News 17(4), 18–23 (2006). [CrossRef]

]. Recent research has mainly focused on minimizing pulse distortion, producing a controllable pulse velocity and delaying a pulse by more than the width of the pulse. With the rapid growth of the transmitting data amount, techniques for controllably delaying ultrafast pulses are urgently demanded in future high speed and large capacity optical networks [5

5. R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Applications of Slow Light in Telecommunications,” Opt. Photon. News 17(4), 18–23 (2006). [CrossRef]

]. Slow light techniques that can offer large fractional delay and distortion-free propagation are especially attractive.

2. Theoretical model

In our work, we consider that a femtosecond pulse with 50 fs duration is incident onto a z-cut 5 mol% MgO: PPLN crystal along y direction (see inset of Fig. 1
Fig. 1 The waveform of the input (black dot line) and the output (red solid line for delay and blue dash line for advancement) pulse. Inset: schematic diagram of the group velocity control scheme through quadratic cascading interaction.
). The signal pulse which is called the FF pulse (central wavelength at ωFF) generates the second harmonic (SH) wave first, then the SH wave will be converted back to the FF due to wave-vector mismatching. When the energy is converted to the SH field, it propagates with the group velocity of the SH field. As a result, the signal experiences deceleration or acceleration since it is dragged by the slower or faster SH pulse depending on the sign of the slight group-velocity mismatch (GVM) and energy exchange between the signal and the SH. We choose 5mol% MgO:PPLN crystal to be the representative medium due to its unique material dispersion [9

9. M. J. Gong, Y. P. Chen, F. Lu, and X. F. Chen, “All optical wavelength broadcast based on simultaneous Type I QPM broadband SFG and SHG in MgO:PPLN,” Opt. Lett. 35(16), 2672–2674 (2010). [CrossRef] [PubMed]

,10

10. J. F. Zhang, Y. P. Chen, F. Lu, and X. F. Chen, “Flexible wavelength conversion via cascaded second order nonlinearity using broadband SHG in MgO-doped PPLN,” Opt. Express 16(10), 6957–6962 (2008). [CrossRef] [PubMed]

,13

13. N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82(20), 3388–3390 (2003). [CrossRef]

]: in type-I geometry (e:o + o), the group velocity matching for second harmonic generation (SHG) occurs at the fundamental wavelength of 1560 nm combined with broadband QPM [10

10. J. F. Zhang, Y. P. Chen, F. Lu, and X. F. Chen, “Flexible wavelength conversion via cascaded second order nonlinearity using broadband SHG in MgO-doped PPLN,” Opt. Express 16(10), 6957–6962 (2008). [CrossRef] [PubMed]

].

In order to get an estimated image for our simulation, we here first perform an approximate analysis in the absence of dispersion. We first consider a type I QPM geometry in this quadratic cascading interaction and the nonlinear coefficient d31 is used. The coupled wave equations governing the propagation of the FF and SH waves under the slowly varying envelope approximation can be generalized as [14

14. G. P. Agrawal, Nonlinear fiber optics, 3rd ed., (Academic Press, 2001).

]:
E1z+ik1''22E1t2=iρ1E1*E2exp(iΔk0z)+iσ1[|E1|2E1+2|E2|2E1]E2z+δE2t+ik2''22E2t2=iρ2E1E1exp(iΔk0z)+iσ2[|E2|2E2+2|E1|2E2].
(1)
where ρi(z) = ± ωid31/cni (the positive and negative sign depend on the distribution of second-order nonlinearity along the QPM grating, and i = 1,2) and σi = 3ωiχ(3)/8cni. The χ(3) terms are the cubic nonlinear terms include self phase modulation (SPM) and cross phase modulation (XPM). Since the input intensity is quite high, they cannot be neglected in our simulation. And they may counteract the cascading quadratic nonlinearity and reduce the net nonlinearity which will contribute to the construction of slow light soliton. Ei(z,t) denotes the amplitude of the electric field, ni denotes the refractive index and ωi denotes the angular frequency. The subscripts 1 and 2 correspond to FF and SH pulses, respectively. Time t is measured in a time frame moving with the linear group velocity of the FF pulse. ki is the inverse group velocity, and ki = d2ki/dω2 is the group velocity dispersion (GVD); δ = kSH−kFF is the GVM, Δk0 = k2-2k1 is derived from Sellmeier’s equation for MgO:LN [15

15. D. E. Zelmon, D. L. Small, and D. Jundt, “Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol. % magnesium oxide-doped lithium niobate,” J. Opt. Soc. Am. B 14(12), 3319–3322 (1997). [CrossRef]

].In our simulation, only the FF light is incident onto the quadratic medium, in the limit of large wave-vector mismatch, an equation of motion for the FF field can be derived from Eq. (1) as following [16

16. C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2): χ(2) cascading,” J. Opt. Soc. Am. B 11(12), 2434–2443 (1994). [CrossRef]

,17

17. J. P. Torres and L. Torner, “Self-splitting of beams into spatial solitons in planar waveguides made of quadratic nonlinear media,” Opt. Quantum Electron. 29(7), 757–776 (1997). [CrossRef]

]:
iE1z+2iδρ1ρ2Δk2|E1|2E1tk2''22E1t2+ρ1ρ2Δk|E1|2E1=0,
(2)
where Δk = k2-2k1-2π/Λ, and Λ is the poling period. In the case of negligible GVD, The general solution E1(z,t) = I1(z,t)1/2exp(iϕ(z,t)) of Eq. (2) can be written as [18

18. F. Baronio, C. De Angelis, M. Marangoni, C. Manzoni, R. Ramponi, and G. Cerullo, “Spectral shift of femtosecond pulses in nonlinear quadratic PPSLT Crystals,” Opt. Express 14(11), 4774–4779 (2006). [CrossRef] [PubMed]

]
I1=f(t+zγI1),
(3)
where I1(z,t) = |E1(z,t)|2 is the square of the amplitude and ϕ(z,t) is the phase
ϕ=zκI1+g(t+zγI1).
(4)
In the Eq. (3) and (4), f(t) = I(t,0) is the initial pulse shape, and g(t) is the initial pulse distribution, γ = −2δρ1ρ2/Δk2, and κ = ρ1ρ2/Δk.

As shown in Eq. (3), the dragging by the slower SH wave in case of γ<0, is proportional to the parameter δ because of increased dragging with larger GVM; And it is also inversely proportional to Δk2, due to the lower amount of SH wave generated from the FF wave and less efficient dragging for increasing wave-vector mismatch. We also noted that the delay of FF pulse increases by increasing the FF pulse intensity, which has detailed discussion in Ref [16

16. C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2): χ(2) cascading,” J. Opt. Soc. Am. B 11(12), 2434–2443 (1994). [CrossRef]

]. by the authors.

3. Results and discussion

The input FF is a transform-limited 50 fs pulse (peak intensity: 50 GW/cm2) at the wavelength of 1530 nm and the temperature is set to be 20°C. Numerical simulation is carried out to solve Eq. (1) with a symmetric split-step beam-propagation method (BPM) and the waveforms of the input and output pulses are shown in Fig. 1. Firstly, we investigate the fractional time delay of the output FF pulse as a function of the FF input peak intensity (as is seen in Fig. 2
Fig. 2 The fractional time delay (blue diamond line) and the quality factor (red circle line) dependences of input pulse peak intensity. The wavelength of the input FF is 1530 nm at the wave-vector mismatch Δkl = 30π. The inset shows the input (blue line) and output (red line) pulse at low input intensity (0.1 GW/cm2).
). Besides, we also evaluate the quality of the output pulses. The pulse shape of quadratic soliton is very close to the hyperbolic-secant or Gaussian functions. In simple terms, a sech2 (t) function is used to fit the central part of the FF and SH pulses at each position. In this paper, the term “Quality Factor” is defined as the fractional amount of energy carried by the central spike of the FF and SH pulses, normalized by the launched energy. It should be noted that, in most of our simulations, when the quality factor drops below 0.75, the pulse will degrade and split into a multi-peaked structure, being no longer tolerable for maintaining pulse integrity. From the inset in Fig. 2, we can see the pulse broadening with no time delay at low intensity case (below 0.1 GW/cm2). Group delay control by varying input FF peak intensity can be realized at higher intensity due to the strong nonlinear interaction and dragging between the FF and SH pulses. The rate of time delay increment decreases with the increasing input peak intensity. This is consistent with the theoretical prediction because the quadratic cascading nonlinearity saturates with the increasing input intensity. But we can also find that with the input pulse peak intensity increased, the quality factor decreases, which is due to the strong interaction between the FF and SH pulses that results in a broad pedestal accompanied by the main spike [19

19. W. J. Lu, Y. P. Chen, L. H. Miu, X. F. Chen, Y. X. Xia, and X. L. Zeng, “All-optical tunable group-velocity control of femtosecond pulse by quadratic nonlinear cascading interactions,” Opt. Express 16(1), 355–361 (2008). [CrossRef] [PubMed]

].

GVM represents the group velocity difference between the FF and the SH pulses. In the presence of GVM, the FF and the SH waves will mutually trap and drag each other and propagate with a velocity between the non-interacting FF and SH group velocities. We can find in Fig. 3(a)
Fig. 3 (a) The fractional time delay (diamond) and the quality factor (circle) as a function of the group velocity mismatch. The blue line shows the case of wave-vector mismatch Δkl = 10π while the red line represents Δkl = 30π. (b) The fractional time delay (diamond) and the quality factor (circle) as a function of the wave-vector mismatch. The wavelength of the input FF is 1530 nm (GVM = 15.5 fs/m) for blue line and 1590 nm (GVM = −14.3 fs/m) for red line.
that the fractional time delay increases nearly linearly with the GVM when it is not too large. It will come to a peak value and then shift to decrease with greater GVM, because the group velocity difference between the FF and SH pulses becomes larger, which counteract the overlap of these two pulses and will weaken the dragging effect. The waveform of the delayed output pulse distorts due to the large walk-off. The quality factor first maintains around a relatively high value and drops to decrease, which is consistent with the tendency of the variation of the fractional time delay. We also found out that, when other circumstances stay the same, smaller wave-vector mismatch (Δk) is always accompanied by lower quality factor and larger fractional time delay.

From above, we can find that with the larger wave-vector mismatch, the group velocity of the FF pulse changes less than that with the smaller mismatch. Then we investigate how the wave-vector mismatch influences over the time delay. From Fig. 3(b) we can find that under the same group velocity mismatch, the time delay slowly decreases towards zero with the wave-vector mismatch increased. The time delay is reduced due to the lower converted amount of the SH waves which contributes to the dragging. The red empty diamond line shows the pulse acceleration case. The smaller wave-vector mismatching can lead to larger fractional time delay but with a lower quality factor. The reason is similar with the explanation given in Fig. 2.

Another factor which may give an important impact on the delaying performance is the input duration of the injected pulses. As can be seen in Fig. 4
Fig. 4 The fractional time delay (blue diamond line) and the quality factor (red circle line) as a function of the input pulse duration. The wavelength and the input peak intensity is 50GW/cm2.
, the fractional time delay come to a peak value around 30 fs where the quality factor is also close to maximum. For it’s the minimum pulse duration that can tolerate the given GVM without the complete walk-off between the FF and SH wave. So in practical case, we can carefully choose the input pulse duration to get a high fractional delay with good pulse quality simultaneously.

Besides these independent analyses above, there’s always a trade-off between the fractional time delay and the quality factor under different situations. It is impossible to obtain infinite time delay without the degradation of the quality factor. So here considering a comprehensive and practical evaluation on the performance of controlling fs laser pulses, we defined an evaluation factor named ‘delay-quality production’ (DQP). The definition equation can be expressed as follows, where τ denotes the fractional time delay and Q is the quality factor.
DQP=τ×Q.
(5)
One can evaluate the delaying performance by DQP. In the practical point of view, since the poling period of the MgO:PPLN crystal is fixed after it is fabricated, we can’t modulate the group velocity mismatch and the wave-vector mismatch independently. It’s more efficient to tune the peak intensity and the duration of the input pulse to obtain the maximum DQP. Figure 5
Fig. 5 DQP as the function of the input pulse duration and the input pulse peak intensity.
gives an example to optimize the delaying performance by selecting the specific peak intensity and duration of the input pulse. We can see that the optimum input pulse duration can always be found with different input peak intensity. This can help us to find the optimum conditions for this group velocity control scheme by looking for the largest achievable DQP.

4. Conclusion

In summary, we presented a full theoretical analysis to evaluate origins of group velocity control of ultrafast pulses through quadratic nonlinear cascading process. Our analyses show that the impact of the fractional time delay depends on the input intensity, group velocity mismatch, wave-vector mismatch and the input pulse duration. In particular we have found that, for a given QPM grating, one can choose the proper peak intensity and pulse duration of the input pulse to obtain the optimum fractional time delay with good pulse quality. According to our simulation results, large time delay with little distortion can be obtained to realize ultrafast optical signal process. Since this slowing down scheme is accompanied with wavelength conversion process, this approach can simultaneously realize these two functions on a single chip which is eagerly demanded for all optical communication systems.

Acknowledgements

References and links

1.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]

2.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301(5630), 200–202 (2003). [CrossRef] [PubMed]

3.

Z. M. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318(5857), 1748–1750 (2007). [CrossRef] [PubMed]

4.

Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, “Large tunable optical delays via self-phase modulation and dispersion,” Opt. Express 14(25), 12022–12027 (2006). [CrossRef] [PubMed]

5.

R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Applications of Slow Light in Telecommunications,” Opt. Photon. News 17(4), 18–23 (2006). [CrossRef]

6.

D. J. Gauthier, “Optical communications - Solitons go slow,” Nat. Photonics 1(2), 92–93 (2007). [CrossRef]

7.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2(11), 775–780 (2006). [CrossRef]

8.

M. Marangoni, C. Manzoni, R. Ramponi, G. Cerullo, F. Baronio, C. De Angelis, and K. Kitamura, “Group-velocity control by quadratic nonlinear interactions,” Opt. Lett. 31(4), 534–536 (2006). [CrossRef] [PubMed]

9.

M. J. Gong, Y. P. Chen, F. Lu, and X. F. Chen, “All optical wavelength broadcast based on simultaneous Type I QPM broadband SFG and SHG in MgO:PPLN,” Opt. Lett. 35(16), 2672–2674 (2010). [CrossRef] [PubMed]

10.

J. F. Zhang, Y. P. Chen, F. Lu, and X. F. Chen, “Flexible wavelength conversion via cascaded second order nonlinearity using broadband SHG in MgO-doped PPLN,” Opt. Express 16(10), 6957–6962 (2008). [CrossRef] [PubMed]

11.

S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78(14), 2752–2755 (1997). [CrossRef]

12.

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278(5339), 843–846 (1997). [CrossRef]

13.

N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82(20), 3388–3390 (2003). [CrossRef]

14.

G. P. Agrawal, Nonlinear fiber optics, 3rd ed., (Academic Press, 2001).

15.

D. E. Zelmon, D. L. Small, and D. Jundt, “Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol. % magnesium oxide-doped lithium niobate,” J. Opt. Soc. Am. B 14(12), 3319–3322 (1997). [CrossRef]

16.

C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2): χ(2) cascading,” J. Opt. Soc. Am. B 11(12), 2434–2443 (1994). [CrossRef]

17.

J. P. Torres and L. Torner, “Self-splitting of beams into spatial solitons in planar waveguides made of quadratic nonlinear media,” Opt. Quantum Electron. 29(7), 757–776 (1997). [CrossRef]

18.

F. Baronio, C. De Angelis, M. Marangoni, C. Manzoni, R. Ramponi, and G. Cerullo, “Spectral shift of femtosecond pulses in nonlinear quadratic PPSLT Crystals,” Opt. Express 14(11), 4774–4779 (2006). [CrossRef] [PubMed]

19.

W. J. Lu, Y. P. Chen, L. H. Miu, X. F. Chen, Y. X. Xia, and X. L. Zeng, “All-optical tunable group-velocity control of femtosecond pulse by quadratic nonlinear cascading interactions,” Opt. Express 16(1), 355–361 (2008). [CrossRef] [PubMed]

20.

W. J. Lu, Y. P. Chen, X. F. Chen, and Y. Xia, “Group Velocity Control of Ultrafast Pulses Based on Electro-Optic Effect and Quadratic Cascading Nonlinearity,” IEEE J. Quantum Electron. 46(7), 1099–1104 (2010). [CrossRef]

21.

M. Conforti, F. Baronio, C. De Angelis, G. Sanna, D. Pierleoni, and P. Bassi, “Spectral shaping of feratosecond pulses in aperiodic quasi-phase-matched gratings,” Opt. Commun. 281, 1693–1697 (2008).

22.

M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, “Observation of spectral drift in engineered quadratic nonlinear media,” Appl. Phys. Lett. 93(2), 021107 (2008). [CrossRef]

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(320.2250) Ultrafast optics : Femtosecond phenomena

ToC Category:
Nonlinear Optics

History
Original Manuscript: December 2, 2010
Revised Manuscript: January 20, 2011
Manuscript Accepted: January 22, 2011
Published: March 4, 2011

Citation
Yu-Ping Chen, Wen-Jie Lu, Yu-Xing Xia, and Xian-Feng Chen, "Dependences of the group velocity for femtosecond pulses in MgO-doped PPLN crystal," Opt. Express 19, 5213-5218 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-6-5213


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References

  1. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]
  2. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301(5630), 200–202 (2003). [CrossRef] [PubMed]
  3. Z. M. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318(5857), 1748–1750 (2007). [CrossRef] [PubMed]
  4. Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, “Large tunable optical delays via self-phase modulation and dispersion,” Opt. Express 14(25), 12022–12027 (2006). [CrossRef] [PubMed]
  5. R. W. Boyd, D. J. Gauthier, and A. L. Gaeta, “Applications of Slow Light in Telecommunications,” Opt. Photon. News 17(4), 18–23 (2006). [CrossRef]
  6. D. J. Gauthier, “Optical communications - Solitons go slow,” Nat. Photonics 1(2), 92–93 (2007). [CrossRef]
  7. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2(11), 775–780 (2006). [CrossRef]
  8. M. Marangoni, C. Manzoni, R. Ramponi, G. Cerullo, F. Baronio, C. De Angelis, and K. Kitamura, “Group-velocity control by quadratic nonlinear interactions,” Opt. Lett. 31(4), 534–536 (2006). [CrossRef] [PubMed]
  9. M. J. Gong, Y. P. Chen, F. Lu, and X. F. Chen, “All optical wavelength broadcast based on simultaneous Type I QPM broadband SFG and SHG in MgO:PPLN,” Opt. Lett. 35(16), 2672–2674 (2010). [CrossRef] [PubMed]
  10. J. F. Zhang, Y. P. Chen, F. Lu, and X. F. Chen, “Flexible wavelength conversion via cascaded second order nonlinearity using broadband SHG in MgO-doped PPLN,” Opt. Express 16(10), 6957–6962 (2008). [CrossRef] [PubMed]
  11. S. N. Zhu, Y. Y. Zhu, Y. Q. Qin, H. F. Wang, C. Z. Ge, and N. B. Ming, “Experimental realization of second harmonic generation in a Fibonacci optical superlattice of LiTaO3,” Phys. Rev. Lett. 78(14), 2752–2755 (1997). [CrossRef]
  12. S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278(5339), 843–846 (1997). [CrossRef]
  13. N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82(20), 3388–3390 (2003). [CrossRef]
  14. G. P. Agrawal, Nonlinear fiber optics, 3rd ed., (Academic Press, 2001).
  15. D. E. Zelmon, D. L. Small, and D. Jundt, “Infrared corrected Sellmeier coefficients for congruently grown lithium niobate and 5 mol. % magnesium oxide-doped lithium niobate,” J. Opt. Soc. Am. B 14(12), 3319–3322 (1997). [CrossRef]
  16. C. R. Menyuk, R. Schiek, and L. Torner, “Solitary waves due to χ(2): χ(2) cascading,” J. Opt. Soc. Am. B 11(12), 2434–2443 (1994). [CrossRef]
  17. J. P. Torres and L. Torner, “Self-splitting of beams into spatial solitons in planar waveguides made of quadratic nonlinear media,” Opt. Quantum Electron. 29(7), 757–776 (1997). [CrossRef]
  18. F. Baronio, C. De Angelis, M. Marangoni, C. Manzoni, R. Ramponi, and G. Cerullo, “Spectral shift of femtosecond pulses in nonlinear quadratic PPSLT Crystals,” Opt. Express 14(11), 4774–4779 (2006). [CrossRef] [PubMed]
  19. W. J. Lu, Y. P. Chen, L. H. Miu, X. F. Chen, Y. X. Xia, and X. L. Zeng, “All-optical tunable group-velocity control of femtosecond pulse by quadratic nonlinear cascading interactions,” Opt. Express 16(1), 355–361 (2008). [CrossRef] [PubMed]
  20. W. J. Lu, Y. P. Chen, X. F. Chen, and Y. Xia, “Group Velocity Control of Ultrafast Pulses Based on Electro-Optic Effect and Quadratic Cascading Nonlinearity,” IEEE J. Quantum Electron. 46(7), 1099–1104 (2010). [CrossRef]
  21. M. Conforti, F. Baronio, C. De Angelis, G. Sanna, D. Pierleoni, and P. Bassi, “Spectral shaping of feratosecond pulses in aperiodic quasi-phase-matched gratings,” Opt. Commun. 281, 1693–1697 (2008).
  22. M. Marangoni, G. Sanna, D. Brida, M. Conforti, G. Cirmi, C. Manzoni, F. Baronio, P. Bassi, C. De Angelis, and G. Cerullo, “Observation of spectral drift in engineered quadratic nonlinear media,” Appl. Phys. Lett. 93(2), 021107 (2008). [CrossRef]

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