## Degenerate four wave mixing in solid core photonic bandgap fibers

Optics Express, Vol. 16, Issue 6, pp. 4059-4068 (2008)

http://dx.doi.org/10.1364/OE.16.004059

Acrobat PDF (782 KB)

### Abstract

Degenerate four wave mixing in solid core photonic bandgap fibers is studied theoretically. We demonstrate the possibility of generating parametric gain across bandgaps, and propose a specific design suited for degenerate four wave mixing when pumping at 532nm. The possibility of tuning the efficiency of the parametric gain by varying the temperature is also considered. The results are verified by numerical simulations of pulse propagation.

© 2008 Optical Society of America

## 1. Introduction

1. P. St. J. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. **24**, 4729–4749 (2006). [CrossRef]

2. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**, 1135 (pages 50) (2006). [CrossRef]

3. R. T. Bise, R. S Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic bandgap fiber,” in Optical Fiber Communications Conference, Post Conference vol. 70 of OSA Trends in Optics and Photonics Series Technical Digest (Optical Society of America, Washington, D.C., 2002), 466–468.

4. T. Larsen, A. Bjarklev, D. Hermann, and J. Broeng, “Optical devices based on liquid crystal photonic bandgap fibres,” Opt. Express **11**, 2589–2596 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-11-20-2589. [CrossRef] [PubMed]

5. F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. **29**, 2369–2371 (2004). [CrossRef] [PubMed]

6. A. Argyros, T. Birks, S. Leon-Saval, C. M. Cordeiro, F. Luan, and P. St. J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express **13**, 309–314 (2005). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-13-1-309. [CrossRef] [PubMed]

7. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. **27**, 1592–1594 (2002). [CrossRef]

*V*-parameter. From the theory of step index fibers we know that a mode is in resonance with the cladding at cut off, so guided modes are expected when the

*V*-parameter is far from a cut off. Therefore in the ARROW-model the guiding properties do not depend on the specific arrangement of the high index cylinders. In spite of its simplicity, the ARROW-model predicts the position of the transmission windows of these fibers with a high accuracy, and therefore fibers guiding light using this principle are also often referred to as ARROW-fibers instead of photonic bandgap fibers. The presence of the resonances in the transmission spectrum of ARROW-fibers gives rise to large normal and anomalous waveguide dispersion for wavelengths near resonance. This effect can be exploited to create fibers with zero dispersion in the blue end of the visible spectrum, and also fibers with multiple zero dispersion points are possible, since every transmission window possesses a zero dispersion frequency. In PCFs with air-hole claddings it requires sub-micron cores and high air fill fractions to achieve zero dispersion wavelengths below 600nm [1

1. P. St. J. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. **24**, 4729–4749 (2006). [CrossRef]

8. K. Saitoh, N. Florous, and M. Koshiba, “Ultra-flattened chromatic dispersion controllability using a defectedcore photonic crystal fiber with low confinement losses,” Opt. Express **13**, 8365–8371 (2005). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-13-21-8365. [CrossRef] [PubMed]

9. K. S. Abedin, J. T. Gopinath, E. P. Ippen, C. E. Kerbage, R. S. Windeler, and B. J. Eggleton, “Highly nondegenerate femtosecond four-wave mixing in tapered microstructure fiber,” Appl. Phys. Lett. **81**, 1384–1386 (2002). [CrossRef]

10. D. A. Akimov, E. E. Serebryannikov, A. M. Zheltikov, M. Schmitt, R. Maksimenka, W. Kiefer, K. V. Dukel’skii, V. S. Shevandin, and Y. N. Kondrat’ev, “Efficient anti-Stokes generation through phase-matched four-wave mixing in higher-order modes of a microstructure fiber,” Opt. Lett. **28**, 1948–1950 (2003). [CrossRef] [PubMed]

11. A. Fuerbach, P. Steinvurzel, J. A. Bolger, A. Nulsen, and B. J. Eggleton, “Nonlinear propagation effects in antiresonant high-index inclusion photonic crystal fibers,” Opt. Lett. **30**, 830–832 (2005). [CrossRef] [PubMed]

12. A. Fuerbach, P. Steinvurzel, J. A. Bolger, and B. J. Eggleton, “Nonlinear pulse propagation at zero dispersion wavelength in anti-resonant photonic crystal fibers,” Opt. Express **13**, 2977–2987 (2005). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-13-8-2977. [CrossRef] [PubMed]

14. I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express **12**, 124–135 (2004). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-12-1-124. [CrossRef] [PubMed]

15. S. Lebrun, P. Delaye, R. Frey, and G. Roosen, “High-efficiency single-mode Raman generation in a liquid-filled photonic bandgap fiber,” Opt. Lett. **32**, 337–339 (2007). [CrossRef] [PubMed]

## 2. Basic theory of degenerate four wave mixing

*ω*

_{1}and

*ω*

_{2}are annihilated, and two new photons at frequencies

*ω*

_{3}and

*ω*

_{4}are generated. When the process is described in the time domain it is usually referred to as modulational instability. Energy conservation requires that

*ω*

_{1}+

*ω*

_{2}=

*ω*

_{3}+

*ω*

_{4}. In order to have the process occurring effectively phase matching is also required. If nonlinearities are neglected the phase mismatch is given by Δ

*β*=

*β*(

*ω*

_{4})+

*β*(

*ω*

_{3})-

*β*(

*ω*

_{2})-

*β*(

*ω*

_{1}), where

*β*(

*ω*) is the propagation constant at the frequency

*ω*. In this work we are mainly interested in the process denoted as

*degenerate four wave mixing*, where

*ω*

_{1}=

*ω*

_{2}=

*ω*. Conservation of energy then requires that the two generated photons have frequencies

_{p}*ω*

_{3}=

*ω*-Ω and

_{p}*ω*

_{4}=

*ω*+Ω, i.e. the generated frequencies are placed symmetrically around the pump frequency

_{p}*ω*. The low frequency component

_{p}*ω*=

_{a}*ω*-Ω, is denoted the Stokes frequency, while the high frequency component

_{p}*ω*=

_{as}*ω*+Ω, is denoted the anti-Stokes frequency. If the nonlinear response is assumed to consist of only an instantaneous Kerr-part, and the mode profile of the guided mode is constant, the parametric gain can be shown to be given by

_{p}*P*

_{0}is the power of the pump,

*κ*=2

*P*

_{0}

*γ*+Δ

*β*is the nonlinear phase mismatch, and

*γ*is the nonlinear parameter given by

*γ*=2

*πn*

_{2}/(

*λ*

_{0}

*A*), where

_{eff}*n*

_{2}is the nonlinear refractive index,

*λ*

_{0}is the pump wavelength, and

*A*is the effective area of the guided mode, here assumed not to vary with wavelength. Therefore, in order to have a gain, we must have

_{eff}*P*

_{0}

*γ*>|

*κ*|/2, which can also be formulated as -4

*P*

_{0}

*γ*<Δ

*β*<0, i.e. the linear phase-shift Δ

*β*must be negative, but greater than -4

*P*

_{0}

*γ*. The highest possible gain is achieved if Δ

*β*=-2

*P*

_{0}

*γ*, where the gain is

*g*=2

_{max}*P*

_{0}

*γ*. When using pulses for FWM the walk-off length is also an important parameter, since the pump, Stokes, and anti-Stokes signals should be overlapping in time to get an efficient gain. The walk-off length is given by

*L*=

_{W}*T*

_{0}|

*ν*

^{-1}

_{gp}-

*ν*

^{-1}

_{g1}|

^{-1}, where

*T*

_{0}is the pulse duration,

*ν*is the group velocity at the pump wavelength, and

_{gp}*v*

_{g1}is the group velocity at the Stokes or anti-Stokes wavelength. The walk-off length is then the physical distance two simultaneously launched pulses with wavelengths

*λ*and

_{p}*λ*

_{1}should travel before their temporal separation is

*T*

_{0}.

## 3. Infiltrated ARROW-fibers

17. Comsol Multiphysics 3.3a (2007) URL www.comsol.com.

*β*, which is strongly dependent on the chromatic dispersion of the fiber, therefore we include both the material dispersion of silica and the high index oils in our calculations. For the material dispersion of silica we use a Sellmeier equation, and for the material dispersion of the high index oils we use Cauchy polynomials that have been fitted to experimentally measured indices of refraction [18

18. Cargille Laboratories, Specifications of Cargille Optical Liquids URL www.cargille.com.

### 3.1. Transmission properties

19. P. Steinvurzel, C. M. de Sterke, B. J. Eggleton, B. T. Kuhlmey, and M. J. Steel, “Mode field distributions in solid core photonic bandgap fibers,” Opt. Commun. **263**, 207–213 (2006). [CrossRef]

**E**

_{λ}(

*x*,

*y*) is the profile of the guided mode at the wavelength

*λ*, and

*n*(

*x*,

*y*) is the refractive index profile of the fiber. The integral in the numerator is taken only over the high index regions, while the integral in the denominator is taken over the whole transverse structure. When dealing with ARROW-fibers

*η*is usually less than 5%, unless the wavelength is lying close to a resonance. The contribution to the linear loss of the fiber due to absorption in the liquid is calculated as

*α*=

_{abs, fiber}*ηα*, where

_{abs,oil}*α*is the linear absorption coefficient of the liquid. The total linear loss of the fiber is estimated by adding the leakage loss and absorption loss, since we neglect losses stemming from absorption in silica. At wavelengths far from resonance the absorption loss is dominating the total linear loss, while at wavelengths close to a resonance the leakage loss adds a significant contribution to the total linear loss.

_{abs,oil}*µ*m and

*d*=1.23

*µ*m. The holes are infiltrated with a high index oil that has

*n*=1.60 at

*λ*=589.3nm [18

18. Cargille Laboratories, Specifications of Cargille Optical Liquids URL www.cargille.com.

*β*(

*ω*) for designs with different relative hole diameters but fixed values of the refractive indices. Now, within the scalar approximation one may derive a scaling law [20

20. T. Birks, D. Bird, T. Hedley, J. Pottage, and P. St. J. Russell, “Scaling laws and vector effects in bandgap-guiding fibres,” Opt. Express **12**, 69–74 (2004). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-12-1-69. [CrossRef] [PubMed]

*β*(

*ω*) curve as the refractive indices are changed. This allows us to approximately account for the frequency-dependent shifts in the refractive indices of silica and infiltration liquids caused by material dispersion, and also to predict results for different infiltration liquids based on a single calculation. In this way, we could efficiently check a range of fiber designs for the possibility of phase-matching between bandgaps, and found several designs where this could be achieved. For the most promising designs we have finally calculated

*β*(

*ω*) using a full-vectorial method with the material dispersion included, to verify that the phase-match was not just an artifact of the scalar approximation. Generally we found that for the process to occur at least one of the three wavelengths involved in the process will experience a loss of several dB/m, but further optimization in order to achieve lower losses still remains unexplored.

*η*as defined in Eq. (1) in the 3rd and 4th. bandgap of the considered ARROW-fiber (Λ=3.1

*µ*m and

*d*=1.23

*µ*m,

*n*≈1.60). We see that in the centers of the bandgaps less than 1% of the field is residing in the high index regions. On the upper

_{Liquid}*x*-axis of the plots in Fig. 1 the

*V*-parameter of the individual high-index rods is shown, which makes it easy to identify the cut-offs between the bandgaps, since these coincide with the zero points of the Bessel functions

*J*

_{0}(

*V*) and

*J*

_{1}(

*V*). The spikes in the 4th. bandgap around 425nm and in the 3rd. bandgap around 510nm come from cut-offs of higher order modes (LP

_{41}and LP

_{31}, respectively). In Fig. 1(b) and (c) the leakage loss and propagation loss is shown. The leakage loss arises because the finite number of holes around the core allows a small fraction of the field to leak out, and could therefore be reduced by increasing the number of holes. We see that the leakage loss is significantly higher in the 4th. bandgap than in the 3rd. This happens because the rod modes couple more strongly in the even-numbered bandgaps than in the odd-numbered bandgaps [21

21. T. A. Birks, F. Luan, G. J. Pearce, A Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express **14**, 5688–5698 (2006). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-14-12-5688. [CrossRef] [PubMed]

*η*is nonzero.

*η*will always increase near a bandgap edge, since the core mode is able to couple to a certain mode of the individual rods here. This is an intrinsic property of ARROW-PCFs, and the losses arising due to this cannot be minimized simply by adding more rings of holes to the structure. In Fig. 1(d) the effective area of the guided mode is plotted. In the same plot we have also shown the effective area of the guided mode for an airhole PCF with the same hole diameter and pitch. We see that around the centers of the bandgaps the area of the mode in the ARROW-fiber is smaller than in the air-hole PCF, meaning that the nonlinearity of the ARROW-fiber can actually be even larger than in the air-hole PCF at certain wavelengths.

### 3.2. Phase-mismatch

*β*is an important parameter when calculating the efficiency of DFWM. In Fig. 2(a) (solid curve) we have plotted Δ

*β*as a function of wavelength for a pump at

*λ*=532nm. The fiber has the same structure as the one considered in Fig. 1. Since we have a parametric gain when the linear phase-mismatch satisfies -4

*P*

_{0}

*γ*<Δ

*β*<0, we see that the fiber has two narrow regions with gain on each side of the pump. For the fiber considered here

*γ*≈0.023(Wm)

^{-1}at 532nm, so for a peak power of 10kW, we have a gain when -920m

^{-1}<Δ

*β*<0. Also in Fig. 2(a) the linear phase-mismatch is shown for two other fibers, where the whole structure (pitch and hole diameter) has been up- or down-scaled by a half percent. We see that these variations changes the linear phase mismatch significantly. We therefore conclude that in order to achieve linear phase-match between two different bandgaps in an ARROW-PCF the tolerances on variations in the structural parameters of the fiber are very low. For state of the art air-hole PCFs it is often stated that the variations along the fiber are about one percent for several meters of fiber [22

22. W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, “Demonstration of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express **10**, 609–613 (2002). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-10-14-609. [PubMed]

23. T. Toyoda and M. Yabe, “The temperature dependence of the refractive indices of fused silica and crystal quartz,” J. Phys. D **16**, L97 (1983). [CrossRef]

*dn*/

*dT*=-4.37·10

^{-4}

*K*

^{-1}. Therefore one can optimize the phase-mismatch by varying the temperature of the fiber. In Fig. 2(b) we show the phase-mismatch for a structure where the temperature is raised a few degrees above room temperature. We see that also small variations in the temperature have a high impact on the linear phase-mismatch. By adjusting the temperature only slightly we can therefore optimize the spectral width of the region with parametric gain. Besides the phase-match criteria, we must also consider the walk-off length, which can be calculated from the group velocity as described in Section 2. In Fig. 2(c) the group velocity is plotted as a function of wavelength for both the considered ARROW-fiber and the similar air-hole PCF. We see that in general there will be a walk-off if we consider effects between two different bandgaps, since group velocity match between two wavelengths lying in different bandgap is only possible if one of the wavelengths is lying close to the bandgap edge. When comparing the group velocity in the ARROW-fiber and the air-hole PCF we see that the group velocity mismatches are similar in magnitude in the considered wavelength spectrum. For a pump at 532nm and the corresponding anti-Stokes wavelength (465nm) we find that the walk-off length for a 10ps pulse is about 20cm. As already mentioned at the beginning of this section, linear losses stemming from absorption in the liquid and leakage of the field can have a detrimental effect on the efficiency of the FWM-process. For the fiber considered here, it is the anti-Stokes wavelength that experiences the greatest linear loss. As seen in Fig. 1(c) the linear loss at the pump, Stokes, and anti-Stokes wavelength are about 0.4dB/m, 0.9dB/m, and 7dB/m, respectively. Hence losses should not be neglected when simulating pulse propagation in these fibers.

^{-3}K, for a single 10ps pulse with a peak power of 10kW propagating through 1.0m of fiber. The characteristic time

*τ*for the heat diffusion can be estimated as

*τ*=(

*ρ*

_{0}

*C*)

*R*

^{2}/

*κ*[24], where

*ρ*

_{0}is the density of the material,

*C*is the heat capacity per unit mass,

*κ*is the thermal conductivity, and

*R*is a characteristic length scale of the structure. Using the parameter values for silica and a radius of 3.1

*µ*m (equal to the pitch in the considered fiber), we find a characteristic time for the heat diffusion of the order of 10

^{-5}s, so if the repetition rate of the pump laser is in the kHz range heating of the oil will only be moderate.

### 3.3. Pulse propagation

25. P. V. Mamyshev and S. V. Chernikov, “Ultrashort-pulse propagation in optical fibers,” Opt. Lett. **15**, 1076 (1990). [CrossRef] [PubMed]

26. J. Laegsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express **15**, 16,110–16,123 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-24-16110. [CrossRef]

2. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**, 1135 (pages 50) (2006). [CrossRef]

_{12}mode of a single high index rod. In this region we include an artificial loss of 100dB/m in the simulations shown here. We have carried out simulations with different losses in the regions between the two bandgaps, and observed that the output spectrum is only weakly dependent on the magnitude of this loss. In Fig. 3(a) we have plotted the group velocity dispersion of the fiber with hole diameter

*d*=1.23

*µ*m and pitch Λ=3.1

*µ*m heated 3°C above room temperature. In the following we will use this fiber for our pulse simulations. In Fig. 3(b) we show the output spectrum for a 10ps (FWHM) Sech-shaped pulse with a peak power of 10kW, and a wavelength of

*λ*

_{0}=532nm. The length of the fiber is 1.5m. From the linear phase-mismatch plotted in Fig. 2(b) we see that Stokes and anti-Stokes peaks are expected around 620nm and 460nm, respectively. The output spectrum in Fig. 3(b) shows that the Stokes and anti-Stokes peaks appear at the predicted positions, which are indicated by the arrows in the top of the plot. We see that especially for the anti-Stokes wavelength the result of the approximate theory in Section 2 agrees well with the result of the full simulation, even though the simulation includes both self-steepening, Raman response, and a wavelength dependent effective area of the guided mode, while the approximate theory only includes an instantaneous non-linear response. For the Stokes wavelength there is a noticeable difference between the wavelength predicted by the approximate theory, and the peak in the output spectrum. A more detailed investigation reveals that the Raman response is responsible for this discrepancy. We demonstrate this by carrying out a simulation without the Raman response. The result of this simulation is also shown in Fig. 3(b) (dashed line). In this case the theoretically predicted Stokes wavelength is consistent with the intensity peak in the simulated spectrum. To understand why the Raman effect changes the output spectrum, we have calculated the energy pr. pulse in the 4th. band-gap as a function of propagation distance, as shown in Fig. 3(e). The total input energy pr. pulse is 113nJ, and the result with and without the Raman effect included is shown. In both cases the energy grows exponentially during the first 20cm of the propagation, which corresponds well with the walk-off distance calculated in Section 3.2. The ratio between the energy in the 4th. bandgap without and with the Raman effect included is expected to evolve approximately as exp(0.36

*P*

_{0}

*γz*), since the Raman effect reduces the instantaneous part of the nonlinear response (and hence the parametric gain) by 18% [16]. With the parameters used here this ratio should be ~10

^{6}after 20cm propagation, which is in reasonable agreement with Fig. 3(e). After 20cm of propagation no further amplification of the anti-Stokes signal takes place, and the energy in the 4th. bandgap starts to decrease exponentially, because of the propagation loss of ~7dB/m at the anti-Stokes wavelength. Without the Raman response the energy decreases all the way out to 1.5m, but with the Raman effect included a cascade of Stokes and anti-Stokes Raman peaks eventually reach the regions with parametric gain, and start acting as a seed for DFWM. For the simulations shown here this happens around 90cm. The appearance of the anti-Stokes Raman peaks happens because of a self-induced phase-matching between the pump and the Stokes Raman peak [27

27. S. Coen, D. A. Wardle, and J. D. Harvey, “Observation of Non-Phase-Matched Parametric Amplification in Resonant Nonlinear Optics,” Phys. Rev. Lett. **89**, 273901 (2002). [CrossRef]

## 4. Conclusion

## References and links

1. | P. St. J. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. |

2. | J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. |

3. | R. T. Bise, R. S Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic bandgap fiber,” in Optical Fiber Communications Conference, Post Conference vol. 70 of OSA Trends in Optics and Photonics Series Technical Digest (Optical Society of America, Washington, D.C., 2002), 466–468. |

4. | T. Larsen, A. Bjarklev, D. Hermann, and J. Broeng, “Optical devices based on liquid crystal photonic bandgap fibres,” Opt. Express |

5. | F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. |

6. | A. Argyros, T. Birks, S. Leon-Saval, C. M. Cordeiro, F. Luan, and P. St. J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express |

7. | N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. |

8. | K. Saitoh, N. Florous, and M. Koshiba, “Ultra-flattened chromatic dispersion controllability using a defectedcore photonic crystal fiber with low confinement losses,” Opt. Express |

9. | K. S. Abedin, J. T. Gopinath, E. P. Ippen, C. E. Kerbage, R. S. Windeler, and B. J. Eggleton, “Highly nondegenerate femtosecond four-wave mixing in tapered microstructure fiber,” Appl. Phys. Lett. |

10. | D. A. Akimov, E. E. Serebryannikov, A. M. Zheltikov, M. Schmitt, R. Maksimenka, W. Kiefer, K. V. Dukel’skii, V. S. Shevandin, and Y. N. Kondrat’ev, “Efficient anti-Stokes generation through phase-matched four-wave mixing in higher-order modes of a microstructure fiber,” Opt. Lett. |

11. | A. Fuerbach, P. Steinvurzel, J. A. Bolger, A. Nulsen, and B. J. Eggleton, “Nonlinear propagation effects in antiresonant high-index inclusion photonic crystal fibers,” Opt. Lett. |

12. | A. Fuerbach, P. Steinvurzel, J. A. Bolger, and B. J. Eggleton, “Nonlinear pulse propagation at zero dispersion wavelength in anti-resonant photonic crystal fibers,” Opt. Express |

13. | A. V. Husakou and J. Herrmann, “Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers,” Phys. Rev. Lett. 87 , 203,901 (2001). |

14. | I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express |

15. | S. Lebrun, P. Delaye, R. Frey, and G. Roosen, “High-efficiency single-mode Raman generation in a liquid-filled photonic bandgap fiber,” Opt. Lett. |

16. | G. P. Agrawal, |

17. | Comsol Multiphysics 3.3a (2007) URL www.comsol.com. |

18. | Cargille Laboratories, Specifications of Cargille Optical Liquids URL www.cargille.com. |

19. | P. Steinvurzel, C. M. de Sterke, B. J. Eggleton, B. T. Kuhlmey, and M. J. Steel, “Mode field distributions in solid core photonic bandgap fibers,” Opt. Commun. |

20. | T. Birks, D. Bird, T. Hedley, J. Pottage, and P. St. J. Russell, “Scaling laws and vector effects in bandgap-guiding fibres,” Opt. Express |

21. | T. A. Birks, F. Luan, G. J. Pearce, A Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express |

22. | W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, “Demonstration of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express |

23. | T. Toyoda and M. Yabe, “The temperature dependence of the refractive indices of fused silica and crystal quartz,” J. Phys. D |

24. | R. W. Boyd, |

25. | P. V. Mamyshev and S. V. Chernikov, “Ultrashort-pulse propagation in optical fibers,” Opt. Lett. |

26. | J. Laegsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express |

27. | S. Coen, D. A. Wardle, and J. D. Harvey, “Observation of Non-Phase-Matched Parametric Amplification in Resonant Nonlinear Optics,” Phys. Rev. Lett. |

**OCIS Codes**

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(060.4005) Fiber optics and optical communications : Microstructured fibers

**ToC Category:**

Photonic Crystal Fibers

**History**

Original Manuscript: January 23, 2008

Revised Manuscript: March 7, 2008

Manuscript Accepted: March 7, 2008

Published: March 11, 2008

**Citation**

Per Dalgaard Rasmussen, Jesper Lægsgaard, and Ole Bang, "Degenerate four wave mixing in solid
core photonic bandgap fibers," Opt. Express **16**, 4059-4068 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-4059

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### References

- P. St. J. Russell, "Photonic-Crystal Fibers," J. Lightwave Technol. 24, 4729-4749 (2006). [CrossRef]
- J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135 (2006). [CrossRef]
- R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, "Tunable photonic bandgap fiber," in Optical Fiber Communications Conference, Post Conference vol. 70 of OSA Trends in Optics and Photonics Series Technical Digest (Optical Society of America, Washington, D.C., 2002), 466-468.
- T. Larsen, A. Bjarklev, D. Hermann, and J. Broeng, "Optical devices based on liquid crystal photonic bandgap fibres," Opt. Express 11, 2589-2596 (2003). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-11-20-2589. [CrossRef] [PubMed]
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