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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 16 — Aug. 8, 2005
  • pp: 6181–6192
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The role of the second zero-dispersion wavelength in generation of supercontinua and bright-bright soliton-pairs across the zero-dispersion wavelength

Michael H. Frosz, Peter Falk, and Ole Bang  »View Author Affiliations


Optics Express, Vol. 13, Issue 16, pp. 6181-6192 (2005)
http://dx.doi.org/10.1364/OPEX.13.006181


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Abstract

Supercontinuum generation with femtosecond pulses in photonic crystal fibers with two zero-dispersion wavelengths (ZDWs) is investigated numerically. The role of the higher ZDW is examined for 5 fiber designs with a nearly constant lower ZDW. It is found that the resulting spectrum is mainly determined by self-phase modulation in the first few mm of fiber, followed by soliton self-frequency shift and amplification of dispersive waves. It is demonstrated how femtosecond soliton pulses can be generated with any desired center wavelength in the 1020–1200 nm range by adjusting the fiber length. Further, the generation of a bright-bright soliton-pair from an initial single red-shifted soliton is found. The soliton-pair has one color in the anomalous dispersion region and the other color in the normal dispersion region, which has not previously been described for bright-bright soliton-pairs.

© 2005 Optical Society of America

1. Introduction

Supercontinuum generation in photonic crystal fibers (PCFs) has attracted much attention in the past few years. One of the reasons is the possibility to fabricate small core sizes (~ 1 μm), leading to the high effective nonlinearities necessary to generate broad spectra. Another reason is the possibility to design the dispersion profile of the PCF, which is very important for particularly parametric processes [1

1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, USA,2001).

, 2

2. J. M. Dudley, L. Provino, N. Grossard, H. Maillotte, R. S. Windeler, B. J. Eggleton, and S. Coen, “Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping,” J. Opt. Soc. Am. B 19, 765–771 (2002). [CrossRef]

, 3

3. N. I. Nikolov, T. Sørensen, O. Bang, and A. Bjarklev, “Improving efficiency of supercontinuum generation in photonic crystal fibers by direct degenerate four-wave mixing,” J. Opt. Soc. Am. B 20, 2329–2337 (2003). [CrossRef]

]. An interesting example of the possibilities offered by this design freedom, is the fabrication of PCFs with two zero-dispersion wavelengths (ZDWs) within the optical spectrum. In standard optical fibers there is only one ZDW in the optical spectrum, typically at ~1300 nm. It is well known that the gain bandwidth for four-wave mixing (FWM) is widest in the vicinity of the ZDW due to phase matching conditions [1

1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, USA,2001).

, 2

2. J. M. Dudley, L. Provino, N. Grossard, H. Maillotte, R. S. Windeler, B. J. Eggleton, and S. Coen, “Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping,” J. Opt. Soc. Am. B 19, 765–771 (2002). [CrossRef]

], and that solitons in the vicinity of the ZDW can amplify dispersive waves in the normal dispersion region (NDR) [4

4. N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995). [CrossRef] [PubMed]

, 5

5. G. Genty, M. Lehtonen, H. Ludvigsen, and M. Kaivola, “Enhanced bandwidth of su-percontinuum generated in microstructured fibers,” Opt. Express 12, 3471–3480 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3471. [CrossRef] [PubMed]

, 6

6. A. Efimov, A. J. Taylor, F. G. Omenetto, A. V. Yulin, N. Y. Joly, F. Biancalana, D. V. Skryabin, J. C. Knight, and P. S. J. Russell, “Time-spectrally-resolved ultrafast nonlinear dynamics in small-core photonic crystal fibers: Experiment and modelling,” Opt. Express 12, 6498http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3471.6507 (2004). [CrossRef] [PubMed]

]. Therefore, PCFs with 2 (or more) ZDWs could prove advantageous for efficient supercontinuum generation and spectral shaping.

The resulting spectrum in ref. [7

7. K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keid-ing, R. Kristiansen, K. P. Hansen, and J. J. Larsen, “Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths,” Opt. Express 12, 1045–1054 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1045. [CrossRef] [PubMed]

]consisted mainly of two separated spectral peaks. Such a spectrum could be used in e.g. differential absorption imaging using optical coherence tomography[10

10. J. M. Schmitt, S. H. Xiang, and K. M. Yung, “Differential absorption imaging with optical coherence tomography,” J. Opt. Soc. Am. A 15, 2288–2296 (1998). [CrossRef]

]. In this context it is advantageous to be able to control the spectral location of the two spectral peaks.

For these reasons, it is of fundamental interest to investigate pulse propagation in PCFs with two ZDWs, to clarify the role of the second ZDW and the mechanisms behind the supercontinuum generation. This could allow for better spectral shaping using appropriate fiber design, and therefore lead to light sources especially suited for particular applications such as optical coherence tomography. In this work, we numerically examine supercontinuum generation in 5 different triangular structure PCFs where we change the separation between the two ZDWs from 165 nm to 870 nm by altering the pitch and hole size of the triangular hole structure [11

11. M. H. Frosz, O. Bang, A. Bjarklev, P. E. Andersen, and J. Broeng, “Supercontinuum Generation in Photonic Crystal Fibers: The Role of the Second Zero Dispersion Wavelength,” presented May 25th 2005, CWC1, at CLEO/QELS 2005, Baltimore, Maryland, USA, 22–27 May 2005.

, 12

12. M. H. Frosz, P. Falk, L. T. Pedersen, O. Bang, and A. Bjarklev, “Supercontinuum generation in untapered and tapered photonic crystal fibers with two zero dispersion wavelengths,” talk #5733–36 presented at SPIE Photonics West, San Jose, California, USA, 22–27 January 2005.

]. We pump close to the nearly constant lower ZDW, which allows us to study the influence of the SSFS, as the soliton can then be formed close to the lower ZDW, followed by a red-shift in frequency towards the higher ZDW, due to SSFS. It is shown that this SSFS can be used to generate wavelength-tunable femtosecond solitons, and we demonstrate a spectral shift of more than 30% of the optical frequency. The location of the higher ZDW can limit the red-shift of the soliton, and is found to determine whether dispersive waves or a soliton pair is formed when the red-shifted soliton approaches the higher ZDW. Specifically, we found that for one of the fiber designs, a bright-bright soliton pair is generated across the ZDW. To the best of our knowledge, this has not been previously described.

2. Numerical simulation of pulse propagation

The pulse propagation is simulated using the split-step Fourier method to solve the generalized nonlinear Schrödinger (NLS) equation [1

1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, USA,2001).

]

Az=im2imβ¯mm!mAtm+[1+iω0t][A(z,t)tR(t′)A(z,tt′)2dt′],
(1)

where A(z,t) is the pulse envelope variation in a retarded time frame t moving with the group velocity of the pump, along the fiber axis z. The dispersion parameters β¯m are estimated from a polynomial fit to the fiber dispersion profile β 2(ω) given by [1

1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, USA,2001).

],

β2(ω)=β¯2+β¯3[ωω0]+12β¯4[ωω0]2+16β¯5[ωω0]3+
(2)

where

β¯m=βm(ω0)=(dmβdωm)ω=ω0,
(3)

and we include up to β¯15 to obtain a good polynomial fit to the dispersion profile over the wavelength range of interest. β(ω) is the mode-propagation constant. The dispersion profile for each fiber is calculated numerically using fully-vectorial plane-wave expansions [13

13. S. G. Johnson and J. D. Joannopoulos, “Block-Iterative Frequency-Domain Methods for Maxwell’s Equations in a Planewave Basis,” Opt. Express 8, 173–190 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]

]. We consider triangular PCFs with pitch ʌ and hole diameter d (see Fig. 1). γ = n 2 ω 0/[cA eff] is the nonlinear parameter, where n 2 = 2.6 · 10-20 m2/W is the nonlinear-index coefficient for silica, ω 0 is the center angular frequency of the pump pulse, c is the speed of light in vacuum, and A eff is the effective core area. R(t) is the Raman response function [1

1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, USA,2001).

].

Contrary to the traditional slowly varying envelope approximation, the model is valid for pulses with spectral widths up to about 1/3 of the carrier frequency [14

14. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989). [CrossRef]

]. The model accounts for SPM, FWM, stimulated Raman scattering (SRS) and self-steepening. We simplify the investigation by assuming a polarization maintaining fiber pumped along one polarization axis. Our model includes the wavelength dependence of the effective core areaA eff(λ) using the more general definition suitable for fibers where some of the field energy may reside in the air-holes [15

15. J. Lægsgaard, N. A. Mortensen, and A. Bjarklev, “Mode areas and field-energy distribution in honeycomb photonic bandgap fibers,” J. Opt. Soc. Am. B 20, 2037–2045 (2003). [CrossRef]

]. For all simulations presented in this work the relative change in the total photon number, a measure of the numerical error which is ideally zero in the absence of loss [14

14. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989). [CrossRef]

], was less than 0.2%. We have implemented the adaptive step-size method outlined by Sinkin et al. [16

16. O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R. Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems,” J. Lightwave Technol. 21(1), 61–68 (2003). [CrossRef]

], since it reduces the total number of Fourier transforms, and thus increases computation speed.

To better understand the pulse dynamics, we consider the pulse simultaneously in the time and spectral domain using spectrograms calculated as [7

7. K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keid-ing, R. Kristiansen, K. P. Hansen, and J. J. Larsen, “Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths,” Opt. Express 12, 1045–1054 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1045. [CrossRef] [PubMed]

]

S(z,t,ω)=eiωt′e[t′t]2α2A(z,t′)dt′2,
(4)

where we have used a window size of α = 16 fs. The spectrogram displays the relative temporal positions of the frequency components of the pulse, and is similar to a cross-correlation frequency-resolved optical gating measurement (X-FROG) [17

17. A. Efimov and A. J. Taylor, “Spectral-temporal dynamics of ultrashort Raman solitons and their role in third-harmonic generation in photonic crystal fibers,” Appl. Phys. B 80, 721–725 (2005).http://dx.doi.org/10.1007/s00340-005-1789-2. [CrossRef]

]. We plot the spectrograms on a logarithmic color scale, normalized to S(0,0, ω 0).

3. The role of the higher ZDW

As shown in Fig. 1 we have found that by modifying the pitch ʌ and hole size d of the triangular hole structure the lower ZDW remains constant at ~780 nm, while the higher ZDW varies between 950 nm and 1650 nm [11

11. M. H. Frosz, O. Bang, A. Bjarklev, P. E. Andersen, and J. Broeng, “Supercontinuum Generation in Photonic Crystal Fibers: The Role of the Second Zero Dispersion Wavelength,” presented May 25th 2005, CWC1, at CLEO/QELS 2005, Baltimore, Maryland, USA, 22–27 May 2005.

, 12

12. M. H. Frosz, P. Falk, L. T. Pedersen, O. Bang, and A. Bjarklev, “Supercontinuum generation in untapered and tapered photonic crystal fibers with two zero dispersion wavelengths,” talk #5733–36 presented at SPIE Photonics West, San Jose, California, USA, 22–27 January 2005.

]. The dispersion profiles of the fiber with ʌ = 1.0 μm and the fiber with ʌ = 1.4 μm are similar to the dispersion profile of the fiber examined by Hilligsøe et al. [7

7. K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keid-ing, R. Kristiansen, K. P. Hansen, and J. J. Larsen, “Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths,” Opt. Express 12, 1045–1054 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1045. [CrossRef] [PubMed]

] and Genty et al. [5

5. G. Genty, M. Lehtonen, H. Ludvigsen, and M. Kaivola, “Enhanced bandwidth of su-percontinuum generated in microstructured fibers,” Opt. Express 12, 3471–3480 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3471. [CrossRef] [PubMed]

], respectively. A eff(λ = 804 nm) is 1.38 μm2 for the fiber with ʌ = 1.0 μm and increases with the pitch ʌ. For the ʌ = 1.4 μm fiber, A eff(λ = 804 nm) is 1.97 μm2.

Fig. 1. Left: Calculated dispersion profiles for 5 triangular PCFs with pitch ʌ and relative air-hole size d/ʌ given in the inset. Right: Wavelength λ DW of dispersive waves vs. the soliton center wavelength λ S. The color labelling is the same in both figures.

To focus this investigation on the influence of the higher ZDW, we have used the same input pulse parameters for all simulations: the pump wavelength is λ 0 = 804 nm, the pulse is Gaussian shaped with an intensity full-width-half-maximum (FWHM) of T FWHM = 13 fs, and the peak power is P 0 = 15 kW. When calculating the spectral average power density, we assume a repetition rate of 80 MHz; the power spectral densities S(λ) presented here are thus normalized so that ∫ S(λ)dλ = P av, where P av is the average pulse power of the input pulse. The used parameters are realizable with commercially available femtosecond lasers. Hilligsøe et al. [7

7. K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keid-ing, R. Kristiansen, K. P. Hansen, and J. J. Larsen, “Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths,” Opt. Express 12, 1045–1054 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1045. [CrossRef] [PubMed]

] used a fiber length of 5 cm. Genty et al. [5

5. G. Genty, M. Lehtonen, H. Ludvigsen, and M. Kaivola, “Enhanced bandwidth of su-percontinuum generated in microstructured fibers,” Opt. Express 12, 3471–3480 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3471. [CrossRef] [PubMed]

] used a fiber length up to 1.5 m, but found that the continuum generation was complete after 50 cm of propagation. We have simulated propagation up to a length of 60 cm, 52 cm, and 60 cm for the fibers with ʌ ≤ 1.2 μm, ʌ = 1.3 μm, and ʌ = 1.4 μm, respectively.

A soliton can transfer energy to a dispersive wave when (1) the soliton and the dispersive wave have equal wave numbers, and (2) a significant part of the soliton spectral power is at the dispersive wave wavelength [4

4. N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995). [CrossRef] [PubMed]

]. First, we estimate the wave number (eigenvalue) k sol of a fundamental soliton with temporal width T sol and carrier frequency ω sol. We assume that the soliton spectrum is narrow enough for k sol to only be slightly perturbed by higher-order dispersion, and neglect this perturbation. One can then use the simple NLS equation

Az=iβ2(ωsol)22At2+iγAA2,
(5)

insert the soliton solution A = √P solsech(t/T sol)exp(ik sol z) and find k sol = |β 2(ω sol)|/[2Tsol2] [1

1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, USA,2001).

].

Since the dispersive wave can be generated far from ω sol, we use the NLS equation with all the higher-order dispersion terms (up to β 15) to estimate the wavenumber k lin of the linear dispersive wave. However, since the dispersive wave initially has neglible power, the nonlinearity can be neglected by setting γ=0:

Az=im2imβm(ωsol)m!mAtm.
(6)

By inserting the expression for a dispersive (linear) wave A lin = √P lin exp{i[k lin z - (ω DW - ω sol)t]} into Eq. (6), we obtain for k sol = k lin

β2(ωsol)2Tsol2=m2βm(ωsol)m![ωDWωsol]m,
(7)

where ω DW is the dispersive wave angular frequency. Since only second-order dispersion is considered for the soliton wavenumber, this equation is, strictly speaking, not valid in the immediate vicinity of the ZDWs. However, as shown below, the predictions given by this equation show agreement with the numerical simulations.

When the pump peak power P 0 is sufficiently high for a higher-order soliton to form, the soliton can break up into N fundamental solitons with different peak powers and temporal widths. The shortest fundamental soliton has the highest peak power, and its width is given by [18

18. J. K. Lucek and K. J. Blow, “Soliton self-frequency shift in telecommunications fiber,” Phys. Rev. A 45, 6666–6674 (1992). [CrossRef] [PubMed]

]

Tsol=T02N1=T02(γP0T02β2(ωsol)|)1,
(8)

where T 0 = T FEHM/1.665. This is inserted into Eq. (7) which is then solved for each fiber dispersion profile. We assume that the fundamental soliton temporal width T sol is unchanged as the soliton is down-shifted in frequency due to SSFS; T sol given by Eq. (8) is therefore calculated using the pump peak power P 0, the input pulse temporal width T 0, and the second-order dispersion β¯2 (β 2(ω sol) = β¯2 at ω sol = (ω 0). The result is shown in Fig. 1, which shows the wavelengths λ DW at which dispersive waves can be amplified, as a function of the soliton wavelength λ s. From this figure we expect that a soliton initially launched at λ 0 = 804 nm will amplify dispersive waves at ~ 600 nm in all the 5 fibers investigated here. Dispersive waves in the infrared are not expected to be amplified in the beginning of the fiber, because there is initially not enough spectral power from the soliton in the infrared [5

5. G. Genty, M. Lehtonen, H. Ludvigsen, and M. Kaivola, “Enhanced bandwidth of su-percontinuum generated in microstructured fibers,” Opt. Express 12, 3471–3480 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3471. [CrossRef] [PubMed]

].

Equation (8) gives the width of the shortest fundamental soliton, which was then used for Fig. 1. We have calculated that the temporally longer fundamental solitons are phase-matched to amplify dispersive waves slightly closer to the soliton wavelength than the shortest soliton. More importantly, the frequency shift per unit propagation length caused by SSFS is smaller, because it scales inversely with T sol [19

19. J. Herrmann and A. Nazarkin, “Soliton self-frequency shift for pulses with a duration less than the period of molecular oscillations,” Opt. Lett. 19, 2065–2067 (1994). [CrossRef] [PubMed]

]. This makes the shortest fundamental soliton move quicker towards the higher ZDW and be the first to amplify dispersive waves. It is therefore more important for the generation of dispersive waves than the longer solitons.

Due to small differences in A eff(ω 0) and β¯2, N varies from 3.8 to 3.3 for the ʌ = 1.1 - 1.4 μm fibers. Compared to these fibers, β¯2 for the ʌ = 1.0 μm fiber is ~ 4 times smaller numerically, giving N = 8.2.

3.1. Propagation up to 6 mm

Fig. 2. Calculated spectrograms up to z = 6 mm in the ʌ = 1.0 μm (left) and the ʌ = 1.1 μm fiber (right). The white horizontal lines indicate the ZDWs. pitch1p0_6mm_dB.avi (0.2 MB), pitch1p1_6mm_dB.avi (0.3 MB). 210 computation points were used.
Fig. 3. Left: power spectra after 6 mm of propagation in the ʌ = 1.0 μm fiber. Blue, solid: full simulation; green, dotted: no delayed Raman response; red, dashed: only dispersion terms are β¯2 and β¯3. Right: phase mismatch κ for degenerate FWM at a peak power P 0 = 15 kW and λ 0 = 804 nm, when β¯2 ,β¯4,…, β¯4 are included (blue, solid) and when only β¯2 is included (red, dashed) in Eq. (9).
κ=2γP0+Ω2β¯2+24!Ω4β¯4+26!Ω6β¯6+
(9)

where Ω is the angular frequency shift from the pump in the FWM process. It is seen in Fig. 3 (right) that in case (b) the phase matched (κ = 0) wavelengths are closer to the pump. Furthermore, the solution to Eq. (7) for case (b) predicts wavenumber match to a dispersive wave at ~ 400 nm, instead of ~ 600 nm. The simulation results for case (a) and (b) are shown in Fig. 3 (left).

For case (a) the red- and blue-shifted peaks are at almost the same location as for the full simulation. This shows that SSFS has neglible influence during the first 6 millimeters of propagation. The results for case (b) show that the red- and blue-shifted peaks are even further away from the pump than in the full simulation. This is contrary of what is expected, if degenerate FWM plays a significant role. We can therefore conclude that FWM is negligible. The blue-shifted peak is more blue-shifted in case (b) than in the full simulation, as predicted from the calculated dispersive wave wavenumber match so amplification of dispersive waves may play a role. The initial spectral broadening is therefore mainly due to SPM, possibly assisted by amplification of dispersive waves.

Fig. 4. Power spectra after 6 mm (left) and 6 cm (right) of propagation. The input spectrum is indicated as a thin black line.

In the ʌ = 1.0 μm fiber, the anomalous dispersion region (ADR) is so narrow that the SPM has rapidly moved most of the pulse energy into the normal dispersion region (NDR) (Fig. 2). In the ʌ = 1.1 μm fiber, the red-shifted peak remains in the ADR and can thus form a soliton. It is seen from Fig. 2 that the soliton starts to amplify dispersive waves at 1600 nm at z ~ 3 mm. The dispersive wave wavelength of ~ 1600 nm is correctly predicted from Fig. 1. We note that the dispersive waves immediately spread temporally, as expected.

The power spectra after 6 mm of propagation in each of the 5 examined fibers are compared in Fig. 4. The dispersive wave generated at ~1600 nm in the ʌ = 1.1 μm fiber has too low power to be seen on the linear scale used in the figure. The comparison shows that in all of the 5 fibers there are two distinct peaks, a red-shifted and a blue-shifted, which arise from SPM, as outlined above. Except for the ʌ = 1.0 μm fiber, in all cases the red-shifted peak is still in the ADR after 6 mm and is able to form solitons.

3.2. Propagation beyond 6 mm

Fig. 5. Left: spectrogram for the 1.2 μm PCF up to z = 60 cm (pitch1p2_60cm_dB.avi, 0.9 MB). Right: spectrogram for the 1.3 μm PCF up to z = 52 cm (pitch1p3_52cm.avi, 0.9 MB). The white horizontal lines indicate the ZDWs.

Soliton-pairs across the ZDW are known in the form of bright-dark and bright-gray soliton pairs made possible by XPM between the solitons [1

1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, USA,2001).

]. However, in our case a bright-bright soliton pair across the ZDW is apparently formed, which has not been previously described. The co-propagation of a sech-pulse in the NDR with a sech-pulse in the ADR was investigated in ref [21

21. C. S. Aparna, S. Kumar, and A. Selvarajan, “Suppression of the soliton frequency shifts by nonlinear pairing of pulses,” Opt. Commun. 131, 267–273 (1996). http://dx.doi.org/10.1016/0030-4018(96)00350-1. [CrossRef]

], but only the propagation of the pulse in the ADR was considered, and the influence of the ADR-pulse on the NDR-pulse was neglected. A bright-bright soliton pair can be formed within the same dispersion region [22

22. V. V. Afansyev, Y. S. Kivshar, V. V. Konotop, and V. N. Serkin, “Dynamics of coupled dark and bright optical solitons,” Opt. Lett. 14, 805–807 (1989). [CrossRef]

], but XPM alone cannot allow a bright-bright soliton pair across the ZDW. We therefore expect that the Raman effect also plays a role in this new observation.

It is known that a region of modulational instability (MI) can exist near the ZDW in the NDR, and that this MI can possibly allow bright solitons to form in the NDR [23

23. F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modula-tional instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994). http://dx.doi.org/10.1016/0030-4018(94)90216-X. [CrossRef]

]. One requirement for the MI region to exist at the angular frequency ω is β 2(ω) + β 4(ω2/12 < 0 [24

24. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. 28, 2225–2227 (2003). [CrossRef] [PubMed]

], where Ω is defined as in Eq.(9). Near the higher ZDW of the ʌ = 1.2 μm fiber we have β 4(ω) < 0, so there does not exist an MI region in the NDR (β 2 < 0), and this can therefore not explain the apparent soliton formation in the NDR.

Fig. 6. A close-up of the spectrogram for the 1.2 μm PCF at z = 60 cm. It is seen that the pulse generated in the normal dispersion region has not changed its width significantly over several centimeters. The white horizontal lines indicate the ZDWs.
Fig. 7. (a) The red-infrared part of the pulse spectrum output from the ʌ = 1.3 μm PCF at various fiber lengths.
(b) Wavelength λ B at which there is group-velocity match to λ A, according to Eq. 11). The black line indicates λ B = λ A.

For the 1.3 εm fiber, the soliton has shifted to ~1200 nm at z = 52 cm. Since this is still far from the higher ZDW, the soliton has not yet lost power to dispersive waves in the infrared (Fig. 5). Similarly, in the 1.4 εm fiber the soliton has shifted to ~1170 nm at z = 60 cm without losing power to infrared dispersive waves. In the simulations with a relatively large distance between the ZDWs (e.g. Fig. 5), we note that the soliton seems to be quite stable: it does not split into multiple solitons. In [5

5. G. Genty, M. Lehtonen, H. Ludvigsen, and M. Kaivola, “Enhanced bandwidth of su-percontinuum generated in microstructured fibers,” Opt. Express 12, 3471–3480 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3471. [CrossRef] [PubMed]

] one can observe immediate splitting into multiple solitons which have roughly equal peak power. We believe this difference is due to our use of a shorter pulse length (13 fs) compared to [5

5. G. Genty, M. Lehtonen, H. Ludvigsen, and M. Kaivola, “Enhanced bandwidth of su-percontinuum generated in microstructured fibers,” Opt. Express 12, 3471–3480 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3471. [CrossRef] [PubMed]

] (200 fs); the soliton number N scales as (P 0 T02)1/2, and is almost 6 times smaller in our case compared to [5

5. G. Genty, M. Lehtonen, H. Ludvigsen, and M. Kaivola, “Enhanced bandwidth of su-percontinuum generated in microstructured fibers,” Opt. Express 12, 3471–3480 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3471. [CrossRef] [PubMed]

]. Furthermore, the soliton number N ~ 3 in the beginning of the fibre, but since the peak power decreases due to SPM combined with group-velocity dispersion after a few millimeters of propagation, N could decrease to ~ 1 so that only one fundamental soliton remains.

A potential application of these fibers could e.g. be the generation of fs pulses in the infrared from an initial pulse at λ = 804 nm. It is further demonstrated in Fig. 7(a) that the SSFS can be used to produce fs pulses with an almost Gaussian spectral shape. By choosing a suitable fiber length, the central wavelength of the Gaussian spectrum can be freely selected in the range ~ 1020–1200 nm. The FWHM of each Gaussian spectrum is on the order of 30 nm. The use of the SSFS to generate wavelength-tunable soliton pulses has previously been demonstrated in tapered PCFs [25

25. X. Liu, C. Xu, W. H. Knox, J. K. Chandalia, B. J. Eggleton, S. G. Kosinski, and R. S. Windeler, “Soliton self-frequency shift in a short tapered air-silica microstructure fiber,” Opt. Lett. 26, 358–360 (2001). [CrossRef]

]. For comparison, in [25

25. X. Liu, C. Xu, W. H. Knox, J. K. Chandalia, B. J. Eggleton, S. G. Kosinski, and R. S. Windeler, “Soliton self-frequency shift in a short tapered air-silica microstructure fiber,” Opt. Lett. 26, 358–360 (2001). [CrossRef]

] a SSFS of 20%of the optical frequency (λ = 1.3 → 1.65 μm) was experimentally demonstrated in a 15 cm long tapered fiber, where our results correspond to a SSFS of more than 30% in 52 cm of untapered fiber.

We expect that for longer fiber lengths, the solitons in the 1.3–1.4 μm PCFs will also continue to red-shift until they are in the vicinity of the higher ZDW, followed by an amplification of dispersive waves in the infrared. We note, however, that fiber losses in the infrared region (OH-absorption and confinement loss) may ultimately limit the spectral extension into the infrared [5

5. G. Genty, M. Lehtonen, H. Ludvigsen, and M. Kaivola, “Enhanced bandwidth of su-percontinuum generated in microstructured fibers,” Opt. Express 12, 3471–3480 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3471. [CrossRef] [PubMed]

, 7

7. K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keid-ing, R. Kristiansen, K. P. Hansen, and J. J. Larsen, “Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths,” Opt. Express 12, 1045–1054 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1045. [CrossRef] [PubMed]

].

3.3. Group-velocity matching condition for soliton-pair across ZDW

In this section we derive an equation describing the condition for group-velocity matching. This is a necessary condition if radiation initially formed from amplification of dispersive waves in the NDR, leads to the formation of a soliton-pair consisting of a soliton in the ADR (at λ = λ A) co-propagating with a ‘soliton’ in the NDR (at λ = λ B).

A necessary condition for the two pulses to be co-propagating is that the group-velocity v g is matched, i.e. 1/v g = β 1(ω A) = β 1(ω B). From the definition of β(ω) [1

1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, USA,2001).

] we have,

β1(ω)=β1(ωA)+β2(ωA)[ωωA]+β3(ωA)2[ωωA]2,
(10)

where β m(ω) is given by Eq. (3) and we only include up to β 3(ω A). The condition β 1(ω A) = β 1(ω B) then gives

Δω=2β2(ωA)β3(ωA),
(11)

4. Conclusions and discussion

We have numerically investigated femtosecond pulse propagation in 5 different photonic crystal fibers with an almost equal lower ZDW at ~ 780 nm and with higher ZDWs ranging from 950 nm to 1650 nm. The fibers are all pumped close to the lower ZDW. Our results show that SPM is dominant in the first ~ 6 mm of the fibers. Contrary to [7

7. K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keid-ing, R. Kristiansen, K. P. Hansen, and J. J. Larsen, “Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths,” Opt. Express 12, 1045–1054 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1045. [CrossRef] [PubMed]

], we found that FWM does not play a part in the initial broadening of the spectrum. In [7

7. K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keid-ing, R. Kristiansen, K. P. Hansen, and J. J. Larsen, “Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths,” Opt. Express 12, 1045–1054 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1045. [CrossRef] [PubMed]

] the influence of FWM was tested by shifting the dispersion of the fiber to normal dispersion everywhere,β 2(ω) → β 2 (ω) + β′, thus removing the possibility of phasematched FWM. A significantly weaker spectral broadening was then observed, and from this it was concluded that FWM plays an important role. However, in the normal dispersion regime SPM and group-velocity dispersion will act together to broaden the pulse temporally significantly faster than in the anomalous dispersion regime [1

1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, USA,2001).

]. Shifting the dispersion to normal dispersion everywhere will therefore weaken the SPM induced spectral broadening, and this could explain the observations made in [7

7. K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keid-ing, R. Kristiansen, K. P. Hansen, and J. J. Larsen, “Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths,” Opt. Express 12, 1045–1054 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1045. [CrossRef] [PubMed]

].

For the fiber with the most narrow ADR, SPM caused most of the power to be moved out of the ADR, and further broadening into the infrared was halted. For fibers with a larger separation between the ZDWs a soliton is formed, which gradually red-shifts along the fiber due to soliton self-frequency shift. The larger the higher ZDW is, the further into the infrared the soliton can red-shift without losing power to dispersive waves above the higher ZDW. We demonstrated how this could be used to generate fs soliton pulses with a center wavelength anywhere between 1020–1200 nm from pulses at 804 nm, corresponding to a shift in optical frequency up to 30%, simply by varying the fiber length. This could potentially find an application in e.g. optical coherence tomography, where it is desired to have a light source with a center wavelength above 1000 nm, for increased penetration in highly scattering tissue such as skin.

We also demonstrated the generation of a bright-bright soliton-pair, from one single red-shifted soliton. This occurs when spectral recoil is sufficiently weak to allow the soliton in the anomalous-dispersion region to red-shift close enough to the higher ZDW so that group-velocity matching to the pulse generated in the normal-dispersion region occurs. The colors of the soliton-pair are in the normal and the anomalous dispersion regime, respectively. This has not previously been observed for bright-bright soliton-pairs.

Acknowledgments

P. Falk acknowledges financial support from the STVF BIOLASE grant No. 26-02-0020. The authors would like to thank Jesper Lægsgaard for his helpful guidance on calculating the dispersion curves. We would also like to thank Anders Bjarklev, Jes Broeng and Peter E. Andersen for fruitful discussions.

References and links

1.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, USA,2001).

2.

J. M. Dudley, L. Provino, N. Grossard, H. Maillotte, R. S. Windeler, B. J. Eggleton, and S. Coen, “Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping,” J. Opt. Soc. Am. B 19, 765–771 (2002). [CrossRef]

3.

N. I. Nikolov, T. Sørensen, O. Bang, and A. Bjarklev, “Improving efficiency of supercontinuum generation in photonic crystal fibers by direct degenerate four-wave mixing,” J. Opt. Soc. Am. B 20, 2329–2337 (2003). [CrossRef]

4.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995). [CrossRef] [PubMed]

5.

G. Genty, M. Lehtonen, H. Ludvigsen, and M. Kaivola, “Enhanced bandwidth of su-percontinuum generated in microstructured fibers,” Opt. Express 12, 3471–3480 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3471. [CrossRef] [PubMed]

6.

A. Efimov, A. J. Taylor, F. G. Omenetto, A. V. Yulin, N. Y. Joly, F. Biancalana, D. V. Skryabin, J. C. Knight, and P. S. J. Russell, “Time-spectrally-resolved ultrafast nonlinear dynamics in small-core photonic crystal fibers: Experiment and modelling,” Opt. Express 12, 6498http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3471.6507 (2004). [CrossRef] [PubMed]

7.

K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keid-ing, R. Kristiansen, K. P. Hansen, and J. J. Larsen, “Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths,” Opt. Express 12, 1045–1054 (2004).http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1045. [CrossRef] [PubMed]

8.

D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, “Soliton self-frequency shift cancellation in photonic crystal fibers,” Science 301, 1705–1708 (2003). [CrossRef] [PubMed]

9.

F. Biancalana, D. V. Skryabin, and A. V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004). http://dx.doi.org/10.1103/PhysRevE.70.016615. [CrossRef]

10.

J. M. Schmitt, S. H. Xiang, and K. M. Yung, “Differential absorption imaging with optical coherence tomography,” J. Opt. Soc. Am. A 15, 2288–2296 (1998). [CrossRef]

11.

M. H. Frosz, O. Bang, A. Bjarklev, P. E. Andersen, and J. Broeng, “Supercontinuum Generation in Photonic Crystal Fibers: The Role of the Second Zero Dispersion Wavelength,” presented May 25th 2005, CWC1, at CLEO/QELS 2005, Baltimore, Maryland, USA, 22–27 May 2005.

12.

M. H. Frosz, P. Falk, L. T. Pedersen, O. Bang, and A. Bjarklev, “Supercontinuum generation in untapered and tapered photonic crystal fibers with two zero dispersion wavelengths,” talk #5733–36 presented at SPIE Photonics West, San Jose, California, USA, 22–27 January 2005.

13.

S. G. Johnson and J. D. Joannopoulos, “Block-Iterative Frequency-Domain Methods for Maxwell’s Equations in a Planewave Basis,” Opt. Express 8, 173–190 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]

14.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989). [CrossRef]

15.

J. Lægsgaard, N. A. Mortensen, and A. Bjarklev, “Mode areas and field-energy distribution in honeycomb photonic bandgap fibers,” J. Opt. Soc. Am. B 20, 2037–2045 (2003). [CrossRef]

16.

O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R. Menyuk, “Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems,” J. Lightwave Technol. 21(1), 61–68 (2003). [CrossRef]

17.

A. Efimov and A. J. Taylor, “Spectral-temporal dynamics of ultrashort Raman solitons and their role in third-harmonic generation in photonic crystal fibers,” Appl. Phys. B 80, 721–725 (2005).http://dx.doi.org/10.1007/s00340-005-1789-2. [CrossRef]

18.

J. K. Lucek and K. J. Blow, “Soliton self-frequency shift in telecommunications fiber,” Phys. Rev. A 45, 6666–6674 (1992). [CrossRef] [PubMed]

19.

J. Herrmann and A. Nazarkin, “Soliton self-frequency shift for pulses with a duration less than the period of molecular oscillations,” Opt. Lett. 19, 2065–2067 (1994). [CrossRef] [PubMed]

20.

G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses,” Opt. Express 12, 4614–4624 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4614. [CrossRef] [PubMed]

21.

C. S. Aparna, S. Kumar, and A. Selvarajan, “Suppression of the soliton frequency shifts by nonlinear pairing of pulses,” Opt. Commun. 131, 267–273 (1996). http://dx.doi.org/10.1016/0030-4018(96)00350-1. [CrossRef]

22.

V. V. Afansyev, Y. S. Kivshar, V. V. Konotop, and V. N. Serkin, “Dynamics of coupled dark and bright optical solitons,” Opt. Lett. 14, 805–807 (1989). [CrossRef]

23.

F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, “Modula-tional instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994). http://dx.doi.org/10.1016/0030-4018(94)90216-X. [CrossRef]

24.

J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. 28, 2225–2227 (2003). [CrossRef] [PubMed]

25.

X. Liu, C. Xu, W. H. Knox, J. K. Chandalia, B. J. Eggleton, S. G. Kosinski, and R. S. Windeler, “Soliton self-frequency shift in a short tapered air-silica microstructure fiber,” Opt. Lett. 26, 358–360 (2001). [CrossRef]

OCIS Codes
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

ToC Category:
Research Papers

History
Original Manuscript: July 6, 2005
Revised Manuscript: July 28, 2005
Published: August 8, 2005

Citation
Michael Frosz, Peter Falk, and Ole Bang, "The role of the second zero-dispersion wavelength in generation of supercontinua and bright-bright soliton-pairs across the zero-dispersion wavelength," Opt. Express 13, 6181-6192 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-16-6181


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References

  1. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, San Diego, CA, USA, 2001).
  2. J. M. Dudley, L. Provino, N. Grossard, H. Maillotte, R. S. Windeler, B. J. Eggleton, and S. Coen, �??Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping,�?? J. Opt. Soc. Am. B 19, 765�??771 (2002). [CrossRef]
  3. N. I. Nikolov, T. Sørensen, O. Bang, and A. Bjarklev, �??Improving efficiency of supercontinuum generation in photonic crystal fibers by direct degenerate four-wave mixing,�?? J. Opt. Soc. Am. B 20, 2329�??2337 (2003). [CrossRef]
  4. N. Akhmediev and M. Karlsson, �??Cherenkov radiation emitted by solitons in optical fibers,�?? Phys. Rev. A 51, 2602�??2607 (1995). [CrossRef] [PubMed]
  5. G. Genty, M. Lehtonen, H. Ludvigsen, and M. Kaivola, �??Enhanced bandwidth of supercontinuum generated in microstructured fibers,�?? Opt. Express 12, 3471�??3480 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3471.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3471.</a> [CrossRef] [PubMed]
  6. A. Efimov, A. J. Taylor, F. G. Omenetto, A. V. Yulin, N. Y. Joly, F. Biancalana, D. V. Skryabin, J. C. Knight, and P. S. J. Russell, �??Time-spectrally-resolved ultrafast nonlinear dynamics in small-core photonic crystal fibers: Experiment and modelling,�?? Opt. Express 12, 6498�??6507 (2004). [CrossRef] [PubMed]
  7. K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keiding, R. Kristiansen, K. P. Hansen, and J. J. Larsen, �??Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths,�?? Opt. Express 12, 1045�??1054 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1045.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1045.</a> [CrossRef] [PubMed]
  8. D. V. Skryabin, F. Luan, J. C. Knight, and P. S. J. Russel, �??Soliton self-frequency shift cancellation in photonic crystal fibers,�?? Science 301, 1705�??1708 (2003). [CrossRef] [PubMed]
  9. F. Biancalana, D. V. Skryabin, and A. V. Yulin, �??Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,�?? Phys. Rev. E 70, 016 615 (2004). <a href="http://dx.doi.org/10.1103/PhysRevE.70.016615">http://dx.doi.org/10.1103/PhysRevE.70.016615.</a> [CrossRef]
  10. J. M. Schmitt, S. H. Xiang, and K. M. Yung, �??Differential absorption imaging with optical coherence tomography,�?? J. Opt. Soc. Am. A 15, 2288�??2296 (1998). [CrossRef]
  11. M. H. Frosz, O. Bang, A. Bjarklev, P. E. Andersen, and J. Broeng, �??Supercontinuum Generation in Photonic Crystal Fibers: The Role of the Second Zero Dispersion Wavelength,�?? presented May 25th 2005, CWC1, at CLEO/QELS 2005, Baltimore, Maryland, USA, 22-27 May 2005.
  12. M. H. Frosz, P. Falk, L. T. Pedersen, O. Bang, and A. Bjarklev, �??Supercontinuum generation in untapered and tapered photonic crystal fibers with two zero dispersion wavelengths,�?? talk #5733-36 presented at SPIE Photonics West, San Jose, California, USA, 22-27 January 2005.
  13. S.G. Johnson and J.D. Joannopoulos, �??Block-Iterative Frequency-Domain Methods for Maxwell�??s Equations in a Planewave Basis,�?? Opt. Express 8, 173�??190 (2001). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a> [CrossRef] [PubMed]
  14. K. J. Blow and D. Wood, �??Theoretical description of transient stimulated Raman scattering in optical fibers,�?? IEEE J. Quantum Electron. 25, 2665�??2673 (1989). [CrossRef]
  15. J. Lægsgaard, N. A. Mortensen, and A. Bjarklev, �??Mode areas and field-energy distribution in honeycomb photonic bandgap fibers,�?? J. Opt. Soc. Am. B 20, 2037�??2045 (2003). [CrossRef]
  16. O. V. Sinkin, R. Holzl¨ohner, J. Zweck, and C. R. Menyuk, �??Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems,�?? J. Lightwave Technol. 21(1), 61�??68 (2003). [CrossRef]
  17. A. Efimov and A. J. Taylor, �??Spectral-temporal dynamics of ultrashort Raman solitons and their role in third-harmonic generation in photonic crystal fibers,�?? Appl. Phys. B 80, 721�??725 (2005). <a href="http://dx.doi.org/10.1007/s00340-005-1789-2">http://dx.doi.org/10.1007/s00340-005-1789-2</a> [CrossRef]
  18. J. K. Lucek and K. J. Blow, �??Soliton self-frequency shift in telecommunications fiber,�?? Phys. Rev. A 45, 6666�??6674 (1992). [CrossRef] [PubMed]
  19. J. Herrmann and A. Nazarkin, �??Soliton self-frequency shift for pulses with a duration less than the period of molecular oscillations,�?? Opt. Lett. 19, 2065�??2067 (1994). [CrossRef] [PubMed]
  20. G. Genty, M. Lehtonen, and H. Ludvigsen, �??Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses,�?? Opt. Express 12, 4614�??4624 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4614.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4614.</a> [CrossRef] [PubMed]
  21. C. S. Aparna, S. Kumar, and A. Selvarajan, �??Suppression of the soliton frequency shifts by nonlinear pairing of pulses,�?? Opt. Commun. 131, 267�??273 (1996). <a href="http://dx.doi.org/10.1016/0030-4018(96)00350-1">http://dx.doi.org/10.1016/0030-4018(96)00350-1</a> [CrossRef]
  22. V. V. Afansyev, Y. S. Kivshar, V. V. Konotop, and V. N. Serkin, �??Dynamics of coupled dark and bright optical solitons,�?? Opt. Lett. 14, 805�??807 (1989). [CrossRef]
  23. F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, �??Modulational instability in optical fibers near the zero dispersion point,�?? Opt. Commun. 108, 60�??64 (1994). <a href="http://dx.doi.org/10.1016/0030-4018(94)90216-X">http://dx.doi.org/10.1016/0030-4018(94)90216-X</a>. [CrossRef]
  24. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. S. Russell, �??Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,�?? Opt. Lett. 28, 2225�??2227 (2003). [CrossRef] [PubMed]
  25. X. Liu, C. Xu, W. H. Knox, J. K. Chandalia, B. J. Eggleton, S. G. Kosinski, and R. S. Windeler, �??Soliton selffrequency shift in a short tapered air-silica microstructure fiber,�?? Opt. Lett. 26, 358�??360 (2001). [CrossRef]

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