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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 11 — May. 29, 2006
  • pp: 4774–4779
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Spectral shift of femtosecond pulses in nonlinear quadratic PPSLT Crystals

F. Baronio, C. De Angelis, M. Marangoni, C. Manzoni, R. Ramponi, and G. Cerullo  »View Author Affiliations


Optics Express, Vol. 14, Issue 11, pp. 4774-4779 (2006)
http://dx.doi.org/10.1364/OE.14.004774


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Abstract

Spectral blue- and red-shifts in a range of 100 nm are achieved by propagating 40 fs pulses with a 70 nm spectrum centered at 1450 nm in a 25-mm-long periodically poled stoichiometric lithium tantalate crystal. We show experimentally that these shifts, originating from a phase-mismatched second harmonic generation process under conditions of strong group-velocity mismatch, can be efficiently controlled by acting on pulse intensity and phase-mismatch.

© 2006 Optical Society of America

1. Introduction

Self-induced effects occurring during propagation of ultra-short light pulses in nonlinear media can be exploited to perform a number of manipulations of their properties in the spatial, temporal and spectral domains [1

1. Y. S. Kivshar and G. P. Agrawal, Optical solitons (Academic press, San Diego, 2003).

]. In particular, self-induced spectral-shifts could be exploited for the realization of devices enabling to tune the carrier frequency of femtosecond pulses with continuity and over a broad range. These devices, that are potentially much simpler than optical parametric amplifiers [2

2. G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum 74, 1–18 (2003). [CrossRef]

], could enhance the performances of sources such as fibre lasers [3

3. J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tunnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, “High-power air-clad large-mode-area photonic crystal fiber laser,” Opt. Express 11, 818–823 (2003). [CrossRef] [PubMed]

], which are very compact and efficient but with a fixed frequency.

In the last decades, many theoretical and experimental studies, performed on ultrashort-pulse propagation in cubic media (i.e., liquids, optical fibers, atomic vapors and semiconductors), showed that the cubic nonlinear self-steepening effect, due to propagation in a medium with an intensity-dependent index of refraction, causes asymmetric self-phase modulation and asymmetric frequency spectra of the propagating pulses [4

4. F. De Martini, C.H. Townes, T.K. Gustafson, and P.L. Kelley, “Self-Steepening of Light Pulses,” Phys. Rev. 164, 312–323 (1967). [CrossRef]

9

9. P. Guerreiro, S. Lee, A. Rodrigues, Y. Hu, E. Wright, S. Najafi, J. Mackenzie, and N. Peyghambarian, “Femtosecond pulse propagation near a two-photon transition in a semiconductor quantum-dot waveguide,” Opt. Lett. 21, 659–661 (1996). [CrossRef] [PubMed]

]. In quadratic media, cascaded processes allow large equivalent cubic nonlinearities to be synthesized [10

10. R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992) [CrossRef] [PubMed]

]. Effective self-phase modulation can be induced on a fundamental frequency (FF) wave by cascading interaction with the generated second harmonic (SH) wave under suitable phasemismatch conditions. In the presence of strong group-velocity-mismatch (GVM) self-phase modulation is accompanied by an effective self-steepening effect, that can lead to shifts of the FF pulse spectrum of controllable sign and magnitude [11

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

,12

12. 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, 757–776 (1997). [CrossRef]

]. A spectral red-shift of 1 nm was first observed in PPLN waveguides by investigating spatial trapping of 4 ps pulses with a 1.5 nm spectrum [13

13. P. Pioger, V. Couderc, L. Lefort, A. Barthelemy, F. Baronio, C. De Angelis, Y. Min, V. Quiring, and W. Sohler, “Spatial trapping of short pulses in Ti-indiffused LiNbO3 waveguides,” Opt. Lett. 27, 2182–2184 (2002). [CrossRef]

,14

14. F. Baronio, A. Barthélémy, S. Carrasco, V. Couderc, C. De Angelis, L. Lefort, Y. Min, P. H Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B 6, S182–S189 (2004). [CrossRef]

], while spectral blue-shifts of 10 nm were achieved in BBO by propagating 120 fs pulses with a 10 nm spectrum centered at 790 nm [15

15. F. Ilday, K. Beckwitt, Y. Chen, H. Lim, and F. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B 21, 376–383 (2004). [CrossRef]

]. In the latter case a deep theoretical investigation of the phenomenon was reported, ascribing spectral shifts to an effective quadratic Raman effect.

In this paper we demonstrate experimentally that, by acting on phase-mismatch and pulse intensity of 40 fs FF pulses, spectral blue- and red-shifts can be efficiently controlled in a range of 100 nm after propagation in a 25-mm-long periodically poled stoichiometric lithium tantalate crystal (PPSLT). The physical origin of spectral shifts is ascribed to an effective self-steepening effect and discussed with the help of a simple analytical model underlining the parameters that govern asymmetric self-phase modulation and spectral shift of the propagating pulses. This model has the advantage of highlighting in an intuitive way the influence of phase-mismatch, intensity, pulse duration and also temporal chirp on the frequency shifts.

2. Theoretical analysis

Let us consider a nonlinear type I interaction of a FF wave and a SH wave travelling in a medium with a large quadratic nonlinearity, in a quasi-phase-matched (QPM) geometry. By assuming first-order QPM, plane waves propagating in z direction and a time frame moving with the linear group velocity of the FF pulse, the equations that govern the interaction are [16

16. C. Balslev Clausen, O. Bang, and Yu. S. Kivshar, “Spatial Solitons and Induced Kerr Effects in Quasi-Phase-Matched Quadratic Media,” Phys. Rev. Lett. 78, 4749–4752 (1997). [CrossRef]

]:

jwzβFF22wt2+χFFvw*ejΔkz=0
jvzjδvtβSH22vt2+χSHw2ejΔkz=0
(1)

where w(z,t) and v(z,t) are the electric field envelopes (in V/m) at the FF and SH waves, respectively; β represents the propagation constant, β’ the inverse group velocity, β” the group velocity dispersion (GVD), δ=β’FF -β’SH; Δk=2βFF - βSH+2π/Λ is the effective wavevector-mismatch, where Λ is the QPM period; χFF=2χ(2)/λnFF and χSH=2χ(2)/λnSH, where χ(2) is the nonlinear coefficient, n is the refractive index, λ the FF wavelength. We consider the usual case where only the FF light is incident on the quadratic medium. In the limit of large phase-mismatch, an equation of motion for the FF field can be derived from (1) [11

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

,12

12. 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, 757–776 (1997). [CrossRef]

]:

jwzβFF22wt2+χFFχSHΔkw2w2jδχFFχSHΔk2w2wt=0
(2)

Equation (2) represents a generalized nonlinear Schrodinger equation [17

17. In literature, the fourth term in Eq. (2) is usually referred as the self-steepening term (see for example Ref. [4–9]).

]. An approximate analysis of Eq. (2) was performed in Ref. 15 in the absence of dispersion. Moreover, exact analytical solutions of Eq. (2) were found in Ref. [4

4. F. De Martini, C.H. Townes, T.K. Gustafson, and P.L. Kelley, “Self-Steepening of Light Pulses,” Phys. Rev. 164, 312–323 (1967). [CrossRef]

9

9. P. Guerreiro, S. Lee, A. Rodrigues, Y. Hu, E. Wright, S. Najafi, J. Mackenzie, and N. Peyghambarian, “Femtosecond pulse propagation near a two-photon transition in a semiconductor quantum-dot waveguide,” Opt. Lett. 21, 659–661 (1996). [CrossRef] [PubMed]

], both in the presence and in the absence of dispersion. It was emphasized that the last term in Eq. (2) is responsible for self-steepening of the pulses, leading to temporal profile distortion and asymmetric frequency spectra. In the case of negligible GVD, the general solution w(z,t)=ρ(z,t)exp(jϕ(z,t)) of Eq. (2) can be written as [8

8. A.S. Rodrigues, M. Santagiustina, and E.M. Wright, “Nonlinear pulse propagation in the vicinity of a two-photon resonance,” Phys. Rev. A 52, 3231–3238 (1995). [CrossRef] [PubMed]

]:

ρ2=f(t+zγρ2)
(3a)
φ=zκρ2+g(t+zγρ2)
(3b)

where f(t)=ρ(t,0)2 is the initial pulse shape, g(t) is the initial phase distribution, γ=2δχFFχSH/Δk2 and κ=χFFχSH/Δk.

The nonlinearly induced asymmetry in pulse amplitude is manifested by the term proportional to z in Eq. (3a). As evident in Eq. (3a), for γ<0 (the usual case since the SH wave is slower than the FF one), both positive and negative mismatch conditions lead to a steepening of the trailing edge of the FF pulse, due to the dragging by the slower SH wave. This effect is directly proportional to the parameter δ because of increased dragging for larger GVM; it is inversely proportional to Δk2, due the lower amount of SH and less efficient dragging for increasing phase-mismatch, and it increases with the FF pulse intensity. The group velocity reduction caused by self-steepening was recently demonstrated experimentally [18

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

].

On the other hand, the nonlinearly induced asymmetry in frequency is manifested by the terms proportional to z in Eq. (3b). In the case of zero initial phase distribution (transform-limited input pulse), the variation of the instantaneous frequency is given by:

δω=ϕt=zκρ2t
(4)

In the case of negligible GVM (δ=0), the pulse propagates undistorted and retains its symmetric shape. Therefore, Eq. (4) results in spectral modifications that are characteristic of self-phase modulation, with frequency shifts of opposite sign and equal magnitude on the leading and trailing edge of the pulse. On the other hand, in the presence of GVM, the asymmetric intensity profile of the pulse induced by self-steepening, unbalances the amount of frequency shift on the leading and trailing edges: in particular, for γ<0 the trailing edge of the pulse is steeper than the leading one and produces the strongest shift, towards the blue for Δk<0, and towards the red for Δk>0, the magnitude of such shift being larger for shorter input pulse-widths. It is worth noting that the relationship between group velocity modification and spectral shift of the FF pulse is quite different with respect to the linear case since the reduction of the FF group velocity occurring for γ<0 can be accompanied either by blue-shift [15

15. F. Ilday, K. Beckwitt, Y. Chen, H. Lim, and F. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B 21, 376–383 (2004). [CrossRef]

,18

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

] or by red-shift [13

13. P. Pioger, V. Couderc, L. Lefort, A. Barthelemy, F. Baronio, C. De Angelis, Y. Min, V. Quiring, and W. Sohler, “Spatial trapping of short pulses in Ti-indiffused LiNbO3 waveguides,” Opt. Lett. 27, 2182–2184 (2002). [CrossRef]

,14

14. F. Baronio, A. Barthélémy, S. Carrasco, V. Couderc, C. De Angelis, L. Lefort, Y. Min, P. H Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B 6, S182–S189 (2004). [CrossRef]

] depending on the sign of phase-mismatch. Finally Eq. (3b) shows that spectral shifts can be strongly affected by the initial pulse chirp.

3. Experimental results

For the experimental observation and efficient control of the spectral phenomena we chose a PPSLT crystal because of its strong GVM, large nonlinearity and high optical damage threshold. The 25-mm-long (L) sample presents poling periods ranging from Λ=17.7 µm to Λ=21µm separated by non-poled regions. At the temperature of 160°C used in the experiments these periods correspond to phase matching wavelengths from 1440 nm to 1560 nm. The availability of different poling periods gave us the possibility of tuning the phase mismatch without changing the pulse spectrum. The near-IR pulses are derived from a non-collinear optical parametric amplifiers (NOPA) [2

2. G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum 74, 1–18 (2003). [CrossRef]

] pumped by the second harmonic of an amplified Ti:sapphire laser system (500 µJ, 150 fs, 1 kHz). The NOPA generates 70 nm nearly transform-limited pulses tunable in the near IR (1300–1500 nm) with widths around 40 fs (FWHM in intensity) and energy up to 1 µJ. The crystal length corresponds to 2.5 times the FF dispersion length, and to more than 200 times the GVM length between the FF and the SH waves arising from the nonlinear interaction. The pulses are focused in the middle of the PPSLT crystal with a spot size of 80 µm, giving a 60-mm confocal parameter, much longer than the crystal. This mild focusing enables to neglect both linear and nonlinear spatial effects. At the output of the sample the frequency spectra of the pulses are detected by a spectrometer.

Experiments were carried out by measuring the intensity dependence of the spectral profiles of the FF output pulse as a function of input pulse intensity for different phase-mismatch conditions. At low-intensity (0.1 GW/cm2), independently of phase-mismatch, the FF output pulse spectrum is indistinguishable from the input one. At negative phase-mismatch values, corresponding to a self-defocusing cascaded nonlinearity, at high enough intensity, spectral blue-shifts of the injected pulses are observed. Typical experimental results obtained injecting FF pulses at 1450 nm with ΔkL=-80π are shown in Fig. 1(a). The blue-shift was observed to increase with the injected power, with a maximum value of about 40 nm at an intensity of 20 GW/cm2. Under these conditions a spectral narrowing of ~20% also occurs. The experimental spectra can not be quantitatively fit using the above described analytical model due to the non-negligible role of dispersion, but they are nicely reproduced by full numerical simulations solving Eqs. (1), as shown in Fig 1 (b). The inset shows the evolution of the FF spectrum during propagation in the nonlinear crystal, demonstrating that the effect saturates after short propagation. A method for circumventing this limitation, based on engineered aperiodically poled crystals, was recently proposed [19

19. K. Beckwitt, F. Ilday, and F. Wise, “Frequency shifting with local nonlinearity management in nonuniformly poled quadratic nonlinear materials,” Opt. Lett. 29, 763–765 (2004). [CrossRef] [PubMed]

].

In the time domain, at low intensity, the FF pulse is broadened due to GVD up to ~170 fs. At higher intensity pulse broadening is partially balanced by the nonlinear phase-shift due to the cascaded second-order interaction, resulting in a ~70 fs pulse-width at 20 GW/cm2, and a group-delay shift induced by self-steepening takes place (see Ref. 18 for details). It is worth noting that in the non-poled region of the crystal, even at high intensity, no spectral modification was observed, thus indicating a negligible contribution by third order susceptibility. The spectral shift is thus a pure self-effect of quadratic interactions.

Fig. 1. Experimental (a) and numerical (b) FF spectra at the output of the crystal: I=0.1 GW/cm2 (dotted line); I=7 GW/cm2 (dash-dotted line); I=20 GW/cm2 (solid line). The phase mismatch is ΔkL=-80π. The inset shows the FF spectral evolution in the λ-z plane at I=20 GW/cm2.

At positive phase-mismatch values, corresponding to a self-focusing cascaded nonlinearity, at high enough intensity, spectral red-shifts of the injected pulses are observed. Fig. 2 shows experimental (a) and numerical (b) results for FF pulses injected at 1490 nm with ΔkL=+80π. For an intensity of 7 GW/cm2 a pronounced red-shift of 40 nm is observed, which is well reproduced by numerical simulations. At higher intensity (20 GW/cm2) an even stronger red-shift occurs (~90 nm), which can not be quantitatively reproduced by the numerical model due to the onset of self-focusing (spatial break-up instabilities) leading to an increase of the local intensity. Moreover, in the time domain the combined effect of positive dispersion and self-focusing cascaded nonlinearity gives rise to pulse broadening and also to pulse break-up at high intensities. These effects need to be considered in applications requiring good temporal quality of femtosecond pulses.

Fig. 2. Experimental (a) and numerical (b) FF spectra at the output of the crystal: I=0.1GW/cm2 (dotted line); I=7 GW/cm2 (dash-dotted line); I=20 GW/cm2 (solid line). The phase mismatch is ΔkL=80π.

Fig. 3. Experimental (symbols) and calculated (lines) FF frequency shifts as a function of injected input intensity for different phase-mismatch conditions: ΔkL=-80π (crosses and solid line), ΔkL=-400π (circles and dashed line), ΔkL=80π (stars and dotted line).

4. Conclusions

In this work we demonstrate experimentally that spectral shifts induced by quadratic processes in a regime of strong phase- and group-velocity-mismatch can be efficiently controlled in sign and magnitude through pulse intensity and phase-mismatch. Spectral red- and blue-shifts of more than 40 nm have been obtained by propagating 40 fs pulses in a 25 mm long PPSLT crystal. The role and the weight of the parameters affecting such shifts is highlighted with the help of a simple analytical model relating the observed nonlinear dynamics to the presence of effective quadratic self-phase modulation and self-steepening. These phenomenona can be exploited for the realization of a simple frequency shifter of femtosecond pulses emitted by non tunable laser sources.

Acknowledgments

The authors wish to acknowledge Prof. Kitamura and co-workers of the National Institute for Materials Science (Tsukuba - Japan) for kindly supplying the substrates. We acknowledge financial support from the project PRIN 2005 “Spectral and temporal control of femtosecond pulses by second order nonlinear processes”.

References and links

1.

Y. S. Kivshar and G. P. Agrawal, Optical solitons (Academic press, San Diego, 2003).

2.

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum 74, 1–18 (2003). [CrossRef]

3.

J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tunnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, “High-power air-clad large-mode-area photonic crystal fiber laser,” Opt. Express 11, 818–823 (2003). [CrossRef] [PubMed]

4.

F. De Martini, C.H. Townes, T.K. Gustafson, and P.L. Kelley, “Self-Steepening of Light Pulses,” Phys. Rev. 164, 312–323 (1967). [CrossRef]

5.

T.K. Gustafson, J.P. Taran, H.A. Haus, J.R. Lifsitz, and P.L. Kelley, “Self-Modulation, Self-Steepening, and Spectral Development of Light in Small-Scale Trapped Filaments,” Phys. Rev. 177, 306–313 (1969). [CrossRef]

6.

D. Grischkowsky, M. Loy, and P. Liao, “Adiabatic following model for two-photon transitions: Nonlinear mixing and pulse propagation,” Phys. Rev. A 12, 2514–2533 (1975). [CrossRef]

7.

D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983). [CrossRef]

8.

A.S. Rodrigues, M. Santagiustina, and E.M. Wright, “Nonlinear pulse propagation in the vicinity of a two-photon resonance,” Phys. Rev. A 52, 3231–3238 (1995). [CrossRef] [PubMed]

9.

P. Guerreiro, S. Lee, A. Rodrigues, Y. Hu, E. Wright, S. Najafi, J. Mackenzie, and N. Peyghambarian, “Femtosecond pulse propagation near a two-photon transition in a semiconductor quantum-dot waveguide,” Opt. Lett. 21, 659–661 (1996). [CrossRef] [PubMed]

10.

R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. Van Stryland, and H. Vanherzeele, “Self-focusing and self-defocusing by cascaded second-order effects in KTP,” Opt. Lett. 17, 28–30 (1992) [CrossRef] [PubMed]

11.

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

12.

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, 757–776 (1997). [CrossRef]

13.

P. Pioger, V. Couderc, L. Lefort, A. Barthelemy, F. Baronio, C. De Angelis, Y. Min, V. Quiring, and W. Sohler, “Spatial trapping of short pulses in Ti-indiffused LiNbO3 waveguides,” Opt. Lett. 27, 2182–2184 (2002). [CrossRef]

14.

F. Baronio, A. Barthélémy, S. Carrasco, V. Couderc, C. De Angelis, L. Lefort, Y. Min, P. H Pioger, V. Quiring, L. Torner, and W. Sohler, “Generation of quadratic spatially trapped beams with short pulsed light,” J. Opt. B 6, S182–S189 (2004). [CrossRef]

15.

F. Ilday, K. Beckwitt, Y. Chen, H. Lim, and F. Wise, “Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes,” J. Opt. Soc. Am. B 21, 376–383 (2004). [CrossRef]

16.

C. Balslev Clausen, O. Bang, and Yu. S. Kivshar, “Spatial Solitons and Induced Kerr Effects in Quasi-Phase-Matched Quadratic Media,” Phys. Rev. Lett. 78, 4749–4752 (1997). [CrossRef]

17.

In literature, the fourth term in Eq. (2) is usually referred as the self-steepening term (see for example Ref. [4–9]).

18.

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

19.

K. Beckwitt, F. Ilday, and F. Wise, “Frequency shifting with local nonlinearity management in nonuniformly poled quadratic nonlinear materials,” Opt. Lett. 29, 763–765 (2004). [CrossRef] [PubMed]

OCIS Codes
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons
(190.5940) Nonlinear optics : Self-action effects
(190.7110) Nonlinear optics : Ultrafast nonlinear optics

ToC Category:
Nonlinear Optics

History
Original Manuscript: February 14, 2006
Revised Manuscript: April 27, 2006
Manuscript Accepted: April 29, 2006
Published: May 29, 2006

Citation
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, 4774-4779 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-11-4774


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References

  1. Y. S. Kivshar and G. P. Agrawal, Optical solitons (Academic press, San Diego, 2003).
  2. G. Cerullo and S. De Silvestri, "Ultrafast optical parametric amplifiers," Rev. Sci. Instrum 74, 1-18 (2003). [CrossRef]
  3. J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tunnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, "High-power air-clad large-mode-area photonic crystal fiber laser," Opt. Express 11, 818-823 (2003). [CrossRef] [PubMed]
  4. F. De Martini, C.H. Townes, T.K. Gustafson, and P.L. Kelley, "Self-steepening of Light Pulses," Phys. Rev. 164, 312-323 (1967). [CrossRef]
  5. T.K. Gustafson, J.P. Taran, H.A. Haus, J.R. Lifsitz, and P.L. Kelley, "Self-Modulation, Self-Steepening, and Spectral Development of Light in Small-Scale Trapped Filaments," Phys. Rev. 177, 306-313 (1969). [CrossRef]
  6. D. Grischkowsky, M. Loy, and P. Liao, "Adiabatic following model for two-photon transitions: Nonlinear mixing and pulse propagation," Phys. Rev. A 12, 2514-2533 (1975). [CrossRef]
  7. D. Anderson and M. Lisak, "Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides," Phys. Rev. A 27, 1393-1398 (1983). [CrossRef]
  8. A.S. Rodrigues, M. Santagiustina, and E.M. Wright, "Nonlinear pulse propagation in the vicinity of a two-photon resonance," Phys. Rev. A 52, 3231-3238 (1995). [CrossRef] [PubMed]
  9. P. Guerreiro, S. Lee, A. Rodrigues, Y. Hu, E. Wright, S. Najafi, J. Mackenzie, and N. Peyghambarian, "Femtosecond pulse propagation near a two-photon transition in a semiconductor quantum-dot waveguide," Opt. Lett. 21, 659-661 (1996). [CrossRef] [PubMed]
  10. R. DeSalvo, D. Hagan, M. Sheik-Bahae, G. Stegeman, E. Van Stryland, and H. Vanherzeele, "Self-focusing and self-defocusing by cascaded second-order effects in KTP," Opt. Lett. 17, 28-30 (1992) [CrossRef] [PubMed]
  11. C. Menyuk, R. Schiek, and L. Torner, "Solitary waves due to χ(2): χ(2) cascading," J. Opt. Soc. Am. B 11, 2434-2443 (1994). [CrossRef]
  12. 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, 757-776 (1997). [CrossRef]
  13. P. Pioger, V. Couderc, L. Lefort, A. Barthelemy, F. Baronio, C. De Angelis, Y. Min, V. Quiring, and W. Sohler, "Spatial trapping of short pulses in Ti-indiffused LiNbO3 waveguides, " Opt. Lett. 27, 2182-2184 (2002). [CrossRef]
  14. F. Ilday, K. Beckwitt, Y. Chen, H. Lim, and F. Wise, "Controllable Raman-like nonlinearities from nonstationary, cascaded quadratic processes, " J. Opt. Soc. Am. B 21, 376-383 (2004). [CrossRef]
  15. C. Balslev Clausen, O. Bang, and Yu. S. Kivshar, "Spatial solitons and induced Kerr effects in quasi-phase-matched Quadratic Media," Phys. Rev. Lett. 78, 4749-4752 (1997). [CrossRef]
  16. In literature, the fourth term in Eq. (2) is usually referred as the self-steepening term (see for example Ref. [4-9]). [CrossRef]
  17. 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, 534-536 (2006).
  18. K. Beckwitt, F. Ilday, and F. Wise, "Frequency shifting with local nonlinearity management in nonuniformly poled quadratic nonlinear materials," Opt. Lett. 29, 763-765 (2004). [CrossRef] [PubMed]

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