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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 6 — Mar. 17, 2008
  • pp: 4059–4068
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Degenerate four wave mixing in solid core photonic bandgap fibers

Per Dalgaard Rasmussen, Jesper Lægsgaard, and Ole Bang  »View Author Affiliations


Optics Express, Vol. 16, Issue 6, pp. 4059-4068 (2008)
http://dx.doi.org/10.1364/OE.16.004059


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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

The invention of the photonic crystal fiber (PCF) has had an immense impact on research in nonlinear fiber optics during the last 10 years. Mainly because of the possibilities of making very small cores, and thus large nonlinearities, and because the dispersion properties of PCFs can be tailored, which allow anomalous group velocity dispersion (GVD) at visible wavelengths [1

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

]. Together these two properties make it possible to study a large variety of nonlinear effects, such as soliton formation, four wave mixing (FWM), and supercontinuum generation [2

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

]. The first PCFs had a cladding, which consisted of air-holes placed in a periodic pattern around a defect site created by omitting an air-hole. The guiding mechanism of such a fiber can be understood intuitively by the concept of total internal reflection, since the solid silica core of the fiber has a higher index of refraction than the surrounding cladding, which is made up of high and low index regions consisting of silica and air, respectively. A fascinating feature of PCFs is that if the refractive index of the material in the holes is raised above the refractive index of silica, the structure can still guide light if certain conditions are satisfied. Such fibers have been constructed in various ways. One way is to infiltrate the air-holes of a standard PCF with a liquid that has a higher index of refraction than silica, this has been demonstrated using both high index oils [3

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.

] and liquid crystals [4

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]

]. Another way of constructing such a fiber is by using the stack and draw technique known from PCFs with air-holes, but where the silica tubes have been replaced by doped silica rods with an index of refraction slightly higher than silica [5

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]

]. The guiding mechanism of this type of fiber can clearly not be explained by the concept of modified total internal reflection. Instead various models have been developed to explain the guiding in these fibers. In one model the guiding is explained by considering the photonic crystal that constitutes the cladding of the fiber. It turns out that even for low index contrast such a photonic crystal possesses bandgaps for out of plane propagation [6

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]

]. If light with a frequency lying in one of these bandgaps is injected into the core of the fiber, it will stay confined to the core due to the photonic bandgap existing in the cladding region. Another way of explaining the guiding of this type of fiber is by using a two dimensional analog of the planar anti-resonant reflecting optical waveguide (ARROW) [7

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]

]. In this model the guidance is explained solely by the resonant properties of a cladding cylinder. Since a cladding cylinder is just an ordinary step index fiber, their properties can conveniently be described using the well known 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]

]. PCFs with air-hole claddings that have several regions of anomalous dispersion separated by regions of normal dispersion can also be produced, but this requires modifications of the standard triangular design where all the cladding holes have the same radii, and the core is defined by omitting a single hole [8

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]

]. Therefore ARROW-PCFs offer a unique testbed for investigation of dispersive properties in the visible spectrum. The price one has to pay for these properties is the presence of high loss regions in the transmission spectrum at frequencies where the core mode is nearly in resonance with a guided modes of the cladding cylinders.

In this work we study degenerate four wave mixing (DFWM) in ARROW-PCFs, and investigate the possibilities of pumping in one bandgap, and obtaining a parametric gain in another bandgap. In standard air-hole PCFs parametric gains at frequencies highly detuned from the pump have also been achieved [9

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

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]

], but in general it is only possible when the pump is either in the anomalous dispersion region, or in the normal dispersion region close to the zero dispersion wavelength. Thus, it is difficult to obtain a parametric gain in air-hole PCFs when pumping below 600nm. Propagation of ultrashort pulses in ARROW-PCFs has been studied earlier both theoretically and experimentally in [11

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]

] and [12

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]

], where it was shown how femto-second pulses propagating close to a zero dispersion wavelength radiate energy in the form of a blueshifting dispersive wave, while the pump pulse is red-shifting due to the Raman effect, a phenomena also known from air-hole PCFs [13

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

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]

]. In another recent work it was shown how Raman generation can be limited to the first Stokes order, by using a photonic bandgap fiber where the higher order Stokes frequencies lie in a low transmission region [15

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]

]. But so far, to the best of our knowledge, neither FWM, or effects between two or more bandgaps in ARROW-PCFs have been studied before.

The rest of this paper is organized as follows, in Section 2 we discuss the basics of DFWM in waveguides. Then in the following section we present fiber designs capable of achieving DFWM across bandgaps, and demonstrate this by numerically simulating propagation of pico-second pulses. Section 4 summarizes the main conclusions.

2. Basic theory of degenerate four wave mixing

Four wave mixing is a third-order nonlinear parametric process. Qualitatively it can be described as a process where two photons with frequencies 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 Δβ=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=ωp. Conservation of energy then requires that the two generated photons have frequencies ω 3=ωp-Ω and ω 4=ωp+Ω, i.e. the generated frequencies are placed symmetrically around the pump frequency ωp. The low frequency component ωa=ωp-Ω, is denoted the Stokes frequency, while the high frequency component ωas=ωp+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 g=2(P0γ)2(κ2)2 [16

16. G. P. Agrawal, Nonlinear Fiber Optics, third edition (Academic Press, San Diego, 2001).

], where P 0 is the power of the pump, κ=2P 0 γβ is the nonlinear phase mismatch, and γ is the nonlinear parameter given by γ=2πn 2/(λ 0 Aeff), where n 2 is the nonlinear refractive index, λ 0 is the pump wavelength, and Aeff 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 P 0 γ>|κ|/2, which can also be formulated as -4P 0 γβ<0, i.e. the linear phase-shift Δβ must be negative, but greater than -4P 0 γ. The highest possible gain is achieved if Δβ=-2P 0 γ, where the gain is gmax=2P 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 LW=T 0|ν -1 gp-ν -1 g1|-1, where T 0 is the pulse duration, νgp is the group velocity at the pump wavelength, and 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 λp and λ 1 should travel before their temporal separation is T 0.

3. Infiltrated ARROW-fibers

The fibers we study all have a triangular structure of high index rods around a defect where a rod has been omitted. To calculate the guided modes and eigenfrequencies we use a commercial finite element tool [17

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

], where leakage losses are estimated by including a cylindrical perfectly matched layer (PML) around the boundary of the fiber. All the fibers considered in the following have 6 rings of holes around the core defect, which is also the case for many commercially available PCFs. We are mainly interested in calculating the phase-mismatch Δβ, 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.

]. The absorption of light propagating in the oil has been measured at 12 different wavelengths from 365nm to 1550nm by the manufacturer, and we interpolate these results to calculate the absorption loss in the fibers.

3.1. Transmission properties

ηλ=liquidEλ(x,y)2n(x,y,λ)dxdySiO2+liquidEλ(x,y)2n(x,y,λ)dxdy,
(1)

where 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 αabs, fiber=ηαabs,oil, where αabs,oil 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.

Fig. 1. Properties of the considered ARROW-fiber with Λ=3.1µm, d=1.23µm, and 6 rings of holes, as a function of wavelength. (a) Modal overlap η between guided mode and high index regions. (b) Leakage loss. (c) Propagation loss, i.e. the sum of the leakage loss and absorption loss. (d) Effective area of the guided mode in the infiltrated fiber, and an air-hole PCF with identical dimensions. Notice that the V-parameter of a high-index rod is shown on the upper x-axis on all the plots.

In the following we propose a design suited for generating a DFWM-band in the 4th. bandgap when pumping at 532nm in the 3rd. bandgap. This design has Λ=3.1µm and d=1.23µm. 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.

]. We have found this design by initially neglecting material dispersion, and calculating 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]

] describing the shift of the 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.

3.2. Phase-mismatch

As described in Section 2 the linear phase mismatch Δβ is an important parameter when calculating the efficiency of DFWM. In Fig. 2(a) (solid curve) we have plotted Δβ 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 -4P 0 γβ<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-1β<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]

]. For the relatively short pieces of fiber we are interested in here, it is therefore fair to assume that these variations are much less than a 1%.

One advantage of the liquid infiltrated fibers over their all solid counterparts is the possibility of changing their properties by varying the temperature slightly. This is possible since many liquids have a negative temperature coefficient, which is more than an order of magnitude higher than the positive temperature coefficient of silica [23

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]

]. The oil we consider here has a temperature coefficient of 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.

Fig. 2. Phase difference as a function of wavelength in the 3rd. and 4th. bandgap when pumping at λ 0=532nm. (a) Three different designs where d/Λ is constant, but the whole structure has been up- and down-scaled by a half percent. (b) Effect of varying the temperature. (c) Group velocity as a function of wavelength for the ARROW-PCF and the similar air-hole PCF.

The part of the linear loss that arises because of absorption in the oil will heat up the oil. To estimate how large this effect will be in the ARROW-PCF, we assume that the absorbed energy is distributed evenly between the 6 holes in the first ring around the core defect. If the absorption loss is 1dB/m, and we assume that the energy absorbed is distributed uniformly along the entire fiber there will be a temperature change of the order of 10-3K, 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

24. R. W. Boyd, Nonlinear Optics, second edition (Academic Press, San Diego, 2003).

], 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 3.1µm (equal to the pitch in the considered fiber), we find a characteristic time for the heat diffusion of the order of 10-5s, 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

Fig. 3. (a) Group velocity dispersion in the 3rd. and 4th. band-gap. The fiber parameters are the same as those used in Fig. 2(b) for the case where the fiber is heated 3°C above room temperature. (b) Pulse spectrum after 1.5m [dashed (black) line: Without the Raman effect included in the simulation, and solid (red) line: With the Raman effect included]. The initial pulse is a 10ps (FWHM) sech-pulse with a peak power of 10kW at 532nm. The two arrows in the top of the plot show the theoretically predicted positions of the Stokes and anti-Stokes wavelengths. (c) and (d): Spectrograms on a logarithmic scale at the propagation distance where the energy in the 4th bandgap is highest without and with the Raman effect, respectively. (e) Integrated energy in the 4th. bandgap pr. pulse as a function of propagation length [dashed (black) line: Raman effect neglected, solid (red) line: Raman effect included].

4. Conclusion

In summary, we have investigated the possibility of obtaining parametric gain across bandgaps in an infiltrated bandgap fiber. A necessary condition for parametric gain is a small linear phase mismatch, and our analysis shows that this can indeed be obtained for wavelengths lying in different bandgaps. We further show that because of the unique dispersion properties of ARROW-fibers, it is possible to achieve small linear phase mismatches, even at the blue edge of the visible spectrum, which is not possible in standard air-hole PCFs unless their cores are extremely small. To confirm the results, we have simulated pulse propagation in such a fiber using a model that includes mode profile dispersion and a wavelength dependent loss, which are two common features of ARROW-fibers not shared by their air-hole equivalents.

In this work we considered a design suited for pumping at 532nm. This was mainly done to illustrate how far down in the spectrum such effects are possible, and it should be emphasized that in principle the system could be optimized to work at any wavelength throughout the visible spectrum, by choosing the appropriate fiber parameters and liquid. We also show how the parametric gain can be changed by varying the temperature of the fiber by only a few degrees. Further, our analysis shows that for experimental realization of parametric gain across bandgaps, it is essential to use a fiber with only marginal structural variations along the fiber axis, since the phase-mismatch is highly dependent on the transverse structure of the fiber.

References and links

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]

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]

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 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]

16.

G. P. Agrawal, Nonlinear Fiber Optics, third edition (Academic Press, San Diego, 2001).

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. 263, 207–213 (2006). [CrossRef]

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]

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]

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]

24.

R. W. Boyd, Nonlinear Optics, second edition (Academic Press, San Diego, 2003).

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]

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]

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

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  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]
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  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]
  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]
  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]
  24. R. W. Boyd, Nonlinear Optics, second edition (Academic Press, San Diego, 2003).
  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 Schr¨odinger equation," Opt. Express 15, 16,110-16,123 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-24-16110. [CrossRef]
  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]

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