Abstract

We observe experimentally, for the first time to our knowledge, the simultaneous emission of two strong conjugate resonant dispersive waves by optical solitons. The effect is observed in a small waveguiding glass feature within the cladding of a Kagome hollow-core photonic crystal fiber. We demonstrate theoretically that the phenomenon is attributed to the unusually high fourth-order dispersion coefficient of the waveguiding feature.

© 2008 Optical Society of America

The generation of a broad supercontinuum (SC) in solid-core photonic crystal fibers (PCFs) is a well-studied and established effect (see [1] and references therein). In the femtosecond regime, that is when pumping the optical fiber with a femtosecond pulse in the anomalous dispersion spectral range, the SC generation process is mainly dominated by solitonic dynamics [1]. In addition to the formation of a high-order soliton in the anomalous dispersion spectral range and its subsequent fission into lower-order solitons, each soliton component is sometimes accompanied by the emission of resonant dispersive waves (RDWs) in a spectral range, where the dispersion can be normal [2]. The latter phenomenon entails the emission of narrowband quasi-linear pulses at specific frequencies and results from a special kind of phase matching (PM) between the optical soliton propagation constant and those of the emitted radiation components [3, 4]. The number of the emitted radiation components per soliton and their frequencies are determined by the zero roots of the underlying PM equation, which depends strongly on the high-order dispersion coefficients. For example, as it was theoretically pointed out by Wai et al. [4] and Akhemediev and Karlsson [3] that there can be only one RDW in the case of a medium that possesses a dominant third-order dispersion (3OD) coefficient β3. In the case where the medium dispersion is dominated by a fourth-order dispersion (4OD) coefficient β4, these authors pointed to the possibility of two RDWs. Prior to this Letter, only the 3OD induced RDW has been experimentally identified in the spectra generated within solid-core PCFs [1, 2].

In this Letter, we observe and explain, for the first time to our knowledge, the simultaneous emission of two conjugate RDWs. The two emissions comprise a strong radiation in the visible spectral range of 400500nm and a weak radiation in the near-infrared range of 1600nm. These radiated components are part of a distinctively different SC generation mechanism from the one previously reported using a conventional solid-core PCF. The new SC is generated by coupling femtosecond pulses into one glass feature within the cladding of a Kagome-lattice hollow-core photonic crystal fiber (HC-PCF) [5, 6], [see Fig. 1B ].

Figure 1 shows the physical structure and summarizes the linear properties of the fiber and the employed waveguiding feature located within the fiber cladding. The fiber is a Kagome-lattice HC-PCF with a pitch of 12.5μm [see Fig. 1A] and a strut-thickness of 550nm [6]. The waveguiding feature is a glass node in the fiber cladding that is located at the intersection of the planes, which forms the tessellating star-of-David cladding geometry. The glass-feature is approximately a 1μm×0.8μm rectangular bridge [Fig. 1B]. Under the appropriate launching conditions, the light is readily guided and confined within the bridge, as is illustrated by the near-field of the light transmitted over an 50cm length of fiber [Fig. 1C]. The rectangular shape confers a strong birefringence to this optical waveguide as illustrated in Fig. 1D. Figure 1D shows the calculated group velocity dispersion (GVD) spectrum for the two polarized modes associated with the fundamental spatial mode. The mode whose polarization is aligned with axis 1 [Fig. 1B] is labeled polarization 1, while polarization 2 corresponds to the mode whose polarization is aligned with axis 2.

Polarization 1 mode shows a stronger curvature in its GVD wavelength variation, and more importantly, the GVD is anomalous over a spectral band ranging from 630  to 1020nm. This gives rise to two zero-GVD wavelengths (ZDW), but the GVD of polarization 2 is maintained as normal. This characteristic differs notably from the GVD dependence of a typical PCF whose core is of similar dimensions [7]. A similar GVD was reported in [8] using a particular PCF, but radiation of dispersive waves mediated by 4OD was observed nor explored [9].

This difference in the dispersion properties of the two polarized modes in the bridge has different implications for each polarization’s nonlinear response. Figure 2 shows the generated spectrum when a waveguiding bridge of an 30cm long fiber is excited by a 100fs laser pulse from a frequency-doubled mode-locked erbium fiber laser emitting at 780nm with a repetition frequency of 80MHz and an average coupled power of 50mW. The spectrum is collected using two different spectrometers: a multichannel photospectrometer for the 300900nm range and an optical spectrum analyzer for the range of 8001800nm. When the laser pulse polarization is aligned with polarization 1, the generated spectrum that reaches the fiber output displays evidence of strong effects owing to soliton formation and fission. This is illustrated in Fig. 2A by the peaks at the red side of the pump soliton [1]. The spectrum exhibits two particularly distinctive peaks that rise simultaneously; one is a strong peak at 470nm (labeled BRDW for blueshifted RDW) and the second is located at 1570nm (labeled RRDW for redshifted RDW). The BRDW has the general features of RDW that have been previously observed and theoretically explained by the effect of 3OD [2]. However, there is no prior report on RRDW, which we will show is in fact the BRDW conjugate resonant dispersive emission from the pump soliton around 780nm. It is worth noting that these two peaks are not present for the case of polarization 2 owing to the absence of solitonic dynamics. This is expected from the normal dispersion of polarization 2 mode, and consequently the generated spectrum is likely to be due to the interplay of self-phase modulation, four-wave mixing, and stimulated Raman scattering [1]. Furthermore, the two experimental spectra indicate that the nonlinearity-induced polarization cross coupling is a relatively weak effect. In light of this we limit the rest of this Letter to the study of polarization 1. To verify the nature of BRDW and RRDW we first numerically simulated the pulse propagation in the bridge and second experimentally examined the spectral location of the conjugate RDW with the laser pump wavelength.

The numerical simulation was carried out by solving the standard generalized nonlinear Schrödinger equation (GNLSE) [7]: izψ+D̂(it)ψ+γN̂(ψ2)ψ=0. Here ψ(z,t) is the complex amplitude of the pulse electric field envelope, z is the propagation distance, t is time, γω0n2cAeff(λ) is the fiber nonlinear coefficient, where n2 is the Kerr coefficient of silica glass, and Aeff(λ) is the effective mode area of the glass bridge whose wavelength dependence is shown in Fig. 3A . In Fig. 3A the fraction of power in the air is also shown so as to distinguish the fraction of the coupled power that experiences the nonlinearity owing to silica. The nonlinear operator is defined as N̂(ψ2)αψ2+(1α)R(tt)ψ(t)2dt, where α is the Kerr/Raman nonlinearity ratio (α=0.82 in silica glass) and R(t) is the Raman response function [7]. The dispersion is considered using the dispersion operator D̂(it)m=2Mβm(it)mm! within the GNLSE, where βm is the mth order dispersion coefficient [7].

The dispersion of the present waveguiding bridge exhibits a characteristically high 4OD contribution relative to 3OD [1]. Figure 3B shows the spectra of β3 and the contribution of 4OD relative to that of 3OD in the dispersion operator, β4Δω4. Here Δω is the frequency shift from the pump-soliton frequency, ω0. The spectra show that in the spectral range of anomalous dispersion, 4OD has a comparable contribution in the dispersion operator to that of 3OD, and the ratio of β4Δω4 to β3 is typically 1 order of magnitude larger than in a solid-core PCF with a similar core size at a wavelength near 800nm [1]. As a consequence, the PM condition D(ωω0)=12γψ02 required for a soliton with central amplitude ψ0 to emit dispersive waves will be fulfilled at more than one optical frequency. This special regime is illustrated in Fig. 3C, where the generated spectrum calculated for a different propagation length is shown. Figure 3C reproduces the main features of the measured spectrum [see Fig. 2A]. This includes the well documented soliton-fission generated lower-order solitons and their subsequent Raman self-frequency shifts [1]. More importantly, the calculated spectrum emulates the experimental spectrum by showing the simultaneous formation, in the very early stage of propagation, of two spectral components around 500 and 1600nm. The evidence of the phenomenon underlying their generation is given in Fig. 3D, which shows that the zero crossings of the PM curve D(ωω0)12γψ02 closely match the wavelengths of the two spectral components. This strongly indicates that the two spectral components are indeed the conjugate RDWs as predicted by Akhemediev and Karlsson [3]. To further corroborate the nature of the experimentally observed radiation components, we measured the dependence of their spectral location on the pump wavelength. This was accomplished by using a tunable mode-locked femtosecond Ti:sapphire oscillator. The pulse duration is 25fs, and the repetition rate is 85MHz. Figure 4 shows the spectral evolution of the pump soliton and the two RDWs as the center wavelength of the laser is changed from 770to840nm. The experimental data are shown by points, while the solid curves represent the theoretical results. The agreement between the two sets of data is excellent, thus providing further evidence on the RDWs nature of the two spectral peaks.

In conclusion, we have experimentally observed, for the first time, the simultaneous generation of two quasi-symmetric peaks of RDWs emitted by solitons in a glass waveguide. Their convenient spectral locations, their polarization, and their favorable temporal properties, which are inherited from the mode-locked input laser pulse, make them excellent candidates for ultrashort-pulsed laser sources.

The authors are grateful to C. Locke, J. McFerran, and M. Frosz for useful discussions. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) (UK) and Welch Foundation (grant A1547).

 

Fig. 1 Physical and linear optical properties of the waveguiding node. (A) Scanning electron microscopy of the HC-PCF. (B) Close-up of the node waveguide. (C) Near-field profile of transmitted light through the node. (D) Calculated GVD of the fundamental mode for both polarizations.

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Fig. 2 Output spectrum after propagation through the node of an 30cm long HC-PCF with an input laser pulse in (A) polarization 1 and (B) polarization 2. The traces are recorded by a multichannel spectrometer for the visible region (left-hand side) and by an optical spectrum analyzer for the near-infrared range (right-hand side).

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Fig. 3 Nonlinear properties of the bridge. (A) Calculated wavelength dependence of the effective area (gray curve and left-hand side axis) and of the power fraction in air (black curve and right-hand side axis). Spectrum of β3 and β4Δω4. (C) Numerical results showing the spectral evolution of a 100fs pulse coupled into the node for different propagation lengths. (D) PM curve; the dashed lines indicate the zero-crossings of the PM curve, which coincide with the two distinctive spectral lines of the generated spectrum.

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Fig. 4 Spectral evolution of the RDWs with pump wavelength. The points are the experimental data and the solid curves are the theoretical results for BRDW (blue curve) and for RRDW (light curve). The dotted curve corresponds to the location of the pump soliton

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1. J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006). [CrossRef]  

2. A. V. Husakou and J. Herrmann, Phys. Rev. Lett. 87, 203901 (2001). [CrossRef]   [PubMed]  

3. N. Akhmediev and M. Karlsson, Phys. Rev. A 51, 2602 (1995). [CrossRef]   [PubMed]  

4. P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, Opt. Lett. 11, 464 (1986). [CrossRef]   [PubMed]  

5. F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, Science 298, 399 (2002). [CrossRef]   [PubMed]  

6. F. Couny, F. Benabid, and P. S. Light, Opt. Lett. 31, 3574 (2006). [CrossRef]   [PubMed]  

7. F. Biancalana, D. V. Skryabin, and A. V. Yulin, Phys. Rev. E 70, 016615 (2004). [CrossRef]  

8. K. M. Hilligsøe, T. Andersen, H. Paulsen, C. Nielsen, K. Mølmer, S. Keiding, R. Kristiansen, K. Hansen, and J. Larsen, Opt. Express 12, 1045 (2004). [CrossRef]   [PubMed]  

9. M. Frosz, P. Falk, and O. Bang, Opt. Express 13, 6181 (2005). [CrossRef]   [PubMed]  

References

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  1. J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
    [Crossref]
  2. A. V. Husakou and J. Herrmann, Phys. Rev. Lett. 87, 203901 (2001).
    [Crossref] [PubMed]
  3. N. Akhmediev and M. Karlsson, Phys. Rev. A 51, 2602 (1995).
    [Crossref] [PubMed]
  4. P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, Opt. Lett. 11, 464 (1986).
    [Crossref] [PubMed]
  5. F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, Science 298, 399 (2002).
    [Crossref] [PubMed]
  6. F. Couny, F. Benabid, and P. S. Light, Opt. Lett. 31, 3574 (2006).
    [Crossref] [PubMed]
  7. F. Biancalana, D. V. Skryabin, and A. V. Yulin, Phys. Rev. E 70, 016615 (2004).
    [Crossref]
  8. K. M. Hilligsøe, T. Andersen, H. Paulsen, C. Nielsen, K. Mølmer, S. Keiding, R. Kristiansen, K. Hansen, and J. Larsen, Opt. Express 12, 1045 (2004).
    [Crossref] [PubMed]
  9. M. Frosz, P. Falk, and O. Bang, Opt. Express 13, 6181 (2005).
    [Crossref] [PubMed]

2006 (2)

J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

F. Couny, F. Benabid, and P. S. Light, Opt. Lett. 31, 3574 (2006).
[Crossref] [PubMed]

2005 (1)

2004 (2)

2002 (1)

F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, Science 298, 399 (2002).
[Crossref] [PubMed]

2001 (1)

A. V. Husakou and J. Herrmann, Phys. Rev. Lett. 87, 203901 (2001).
[Crossref] [PubMed]

1995 (1)

N. Akhmediev and M. Karlsson, Phys. Rev. A 51, 2602 (1995).
[Crossref] [PubMed]

1986 (1)

Akhmediev, N.

N. Akhmediev and M. Karlsson, Phys. Rev. A 51, 2602 (1995).
[Crossref] [PubMed]

Andersen, T.

Antonopoulos, G.

F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, Science 298, 399 (2002).
[Crossref] [PubMed]

Bang, O.

Benabid, F.

F. Couny, F. Benabid, and P. S. Light, Opt. Lett. 31, 3574 (2006).
[Crossref] [PubMed]

F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, Science 298, 399 (2002).
[Crossref] [PubMed]

Biancalana, F.

F. Biancalana, D. V. Skryabin, and A. V. Yulin, Phys. Rev. E 70, 016615 (2004).
[Crossref]

Chen, H. H.

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Couny, F.

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Falk, P.

Frosz, M.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Hansen, K.

Herrmann, J.

A. V. Husakou and J. Herrmann, Phys. Rev. Lett. 87, 203901 (2001).
[Crossref] [PubMed]

Hilligsøe, K. M.

Husakou, A. V.

A. V. Husakou and J. Herrmann, Phys. Rev. Lett. 87, 203901 (2001).
[Crossref] [PubMed]

Karlsson, M.

N. Akhmediev and M. Karlsson, Phys. Rev. A 51, 2602 (1995).
[Crossref] [PubMed]

Keiding, S.

Knight, J. C.

F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, Science 298, 399 (2002).
[Crossref] [PubMed]

Kristiansen, R.

Larsen, J.

Lee, Y. C.

Light, P. S.

Menyuk, C. R.

Mølmer, K.

Nielsen, C.

Paulsen, H.

Russell, P. S. J.

F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, Science 298, 399 (2002).
[Crossref] [PubMed]

Skryabin, D. V.

F. Biancalana, D. V. Skryabin, and A. V. Yulin, Phys. Rev. E 70, 016615 (2004).
[Crossref]

Wai, P. K. A.

Yulin, A. V.

F. Biancalana, D. V. Skryabin, and A. V. Yulin, Phys. Rev. E 70, 016615 (2004).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (1)

N. Akhmediev and M. Karlsson, Phys. Rev. A 51, 2602 (1995).
[Crossref] [PubMed]

Phys. Rev. E (1)

F. Biancalana, D. V. Skryabin, and A. V. Yulin, Phys. Rev. E 70, 016615 (2004).
[Crossref]

Phys. Rev. Lett. (1)

A. V. Husakou and J. Herrmann, Phys. Rev. Lett. 87, 203901 (2001).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[Crossref]

Science (1)

F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, Science 298, 399 (2002).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 Physical and linear optical properties of the waveguiding node. (A) Scanning electron microscopy of the HC-PCF. (B) Close-up of the node waveguide. (C) Near-field profile of transmitted light through the node. (D) Calculated GVD of the fundamental mode for both polarizations.
Fig. 2
Fig. 2 Output spectrum after propagation through the node of an 30 cm long HC-PCF with an input laser pulse in (A) polarization 1 and (B) polarization 2. The traces are recorded by a multichannel spectrometer for the visible region (left-hand side) and by an optical spectrum analyzer for the near-infrared range (right-hand side).
Fig. 3
Fig. 3 Nonlinear properties of the bridge. (A) Calculated wavelength dependence of the effective area (gray curve and left-hand side axis) and of the power fraction in air (black curve and right-hand side axis). Spectrum of β 3 and β 4 Δ ω 4 . (C) Numerical results showing the spectral evolution of a 100 fs pulse coupled into the node for different propagation lengths. (D) PM curve; the dashed lines indicate the zero-crossings of the PM curve, which coincide with the two distinctive spectral lines of the generated spectrum.
Fig. 4
Fig. 4 Spectral evolution of the RDWs with pump wavelength. The points are the experimental data and the solid curves are the theoretical results for BRDW (blue curve) and for RRDW (light curve). The dotted curve corresponds to the location of the pump soliton

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