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We report on a possible phase matching between two fundamental modes guided in an appropriately designed photonic crystal fiber. The phase index matching condition can be perfectly fulfilled for second or third harmonic generation and for wavelengths over a large spectral range, simply by tuning the lattice pitch. This can be achieved in such a structure thanks to the coexistence of total internal reflection and photonic bandgap guidance, leading to two different dispersive behaviours for the fundamental and the harmonic waves.
©2008 Optical Society of America
1. Introduction
Since its first proposal[1, 2], photonic crystals have stimulated much research due to their ability to prohibit the propagation of light for certain frequency bands and directions. By introducing a defect in a periodic dielectric structure[3, 4, 5], one can guide light in this defect by a photonic band-gap effect instead of the more usual total internal reflection guidance[6]. A typical example is the development of bandgap guiding Photonic Crystal Fibers (PCF) composed of a guiding low index core surrounding by a periodic dielectric cladding[7, 8, 9]. These particular PCFs exhibit unique spectral and dispersive properties. Conversely, in a high index core PCF, light is guided by modified total internal reflection. These index guiding PCFs have stimulated much research on nonlinear optics as they allow not only to reach very strong light confinement[10] (thanks to a high index contrast between air and silica) but also to obtain unique chromatic Group Velocity Dispersion (GVD) properties (anomalous GVD in the visible domain[11], very flat GVD[12]). A typical breakthrough due to PCFs is the development of compact, powerful and very broadband supercontinua sources[13]. Among all the nonlinear effects, we will focus here on the possibility to fulfil the phase index matching condition for efficient Second Harmonic Generation (SHG) or Third Harmonic Generation (THG) in a hybrid guiding PCF.
Nonlinear effects in photonics are useful to generate new frequencies, leading for example to an extension of the spectral range of laser sources. Despites the low third order susceptibility of silica or the centrosymmetry of the glass, efficient harmonic generation could be expected if phase index matching condition is fulfilled, as long interaction lengths and strong light confinement are available in fibers. Further, note that the centrosymmetry of silica glass can be broken using poling technique in order to induce the required second-order nonlinear coefficient[14].
Phase index matched harmonic generation is not easily obtained in optical fibers as both material and waveguide chromatic effective index dispersions are decreasing functions of the wavelength (in the normal dispersion regime). One could use the anomalous dispersion regions of the material but it is typically accompanied by very high absorption, making this method unattractive. Thus, phase index matching implies to find a waveguide whose dispersion can counterbalance the material dispersion. In bulk silica glass, the index difference Δn material to compensate for SHG and for THG is plotted on Fig. 1. The minimal Δn material is about - 9.5×10-3 for SHG (with a fundamental wavelength λ fund of 1.4 µm) and -17×10-3 for THG (with λ fund =1.7 µm). Thus so far, one uses either the quasi phase-matched technique[15, 16] or the usual phase matching technique consisting in working with two different order guided modes[17]. However, these two solutions have significant drawbacks: the first one is in practical limited to relatively short (cm range) fiber device (as the fabrication of an accurate grating is required) while the second one leads to a low modal overlap and to an intensity profile for the harmonic wave which is not easily exploitable (as the wave is generated in a high order mode). Note that the birefringence phase index matching technique, widely used with nonlinear crystals, cannot be used for fibers since the maximal birefringence that can be experimentally achieved is too low[18].
Fig. 1. Index difference Δnmaterial in bulk silica glass between a fundamental wavelength λ fund and its second or third harmonic λ harm. Δnmaterial=n(λ fund)-n(λ harm).
In this paper, we discuss a fiber structure suitable to fulfil the phase index matching condition between two fundamental modes in order to generate either second harmonic or third harmonic wave with an intensity profile suitable for a direct output coupling. First, we briefly describe this particular fiber and explain why this hybrid guiding PCF proves to be a good candidate for harmonic generation. Then, we describe basic properties of such fiber in the case of few designs and finally we show the large tunability of the phase index matching condition of our structure.
2. Description and particularities of the fiber structure
A classical all-solid bandgap guiding PCF is made of a low index defect core (pure silica) surrounded by a lattice of high index inclusions embedded in a silica background[9]. The typical transmission spectrum of such a fiber exhibits a succession of high and low losses regions, the spectral regions of low losses corresponding to the existence of photonic bandgaps in the periodical cladding. Recently, we have demonstrated that the guidance properties of this kind of fiber can be strongly modified by adding interstitial air holes between the cladding rods[19, 20] (see inset on Fig. 2). The impact of these air holes on the cladding properties is shown on the photonic bandgap diagrams plotted on Fig. 2. Such diagrams exhibit the ranges of wavelengths and effective indices where the propagation of light in the infinite cladding is either allowed (Bloch’s modes) or forbidden (bandgaps). The band diagrams have been calculated thanks to the plane wave expansion method[21] with no material dispersion (background refractive index fixed to 1.45). The opto-geometrical parameters used for all further calculations are fixed, except for the air hole radius, r: the high index rods exhibit a parabolic refractive index profile with a maximal index difference Δn equal to 32×10-3 and with a ratio d/Λ=0.725 where d and Λ are respectively the inclusion diameter and the lattice pitch. In fig. 2, two key features, with a detailed explanation given in [20], has to be pointed out. By adding these air holes, the bandgaps lower boundaries are significantly lowered, all the more as the air holes size is large, (while the upper boundaries are almost not affected) leading to larger and deeper bandgaps. Another important consequence of the air holes insertion concerns the effective index of the first infinite cladding mode (also called Fundamental Space filling Mode index, n FSM). Indeed, as the cladding contains now low index regions, nFSM decreases and can pass through the silica index line (1.45) for long wavelengths and for large enough air holes. As a consequence, in such an index structure, the effective index of the bandgap guided modes proves to be much lower thanks to larger and deeper bandgaps, and the effective index of the fundamental space filling mode nFSM can be tailored to enable a Modified Total Internal Reflection (MTIR) guidance for wavelengths beyong whose corresponding to the bandgaps[20].
Fig. 2. photonic bandgap diagrams of the hybrid structure for different values of r/Λ, calculated without the material dispersion. For each value, the borders of the 3 first bandgaps, as well as the Fundamental Space filling Mode line are plotted. Inset, the cladding unit cell inside the parallelogram.
3. Original application: phase matching for harmonic generation
Figure 3 represents the band diagram of an example of such hybrid structures allowing MTIR and bandgap guidance. The effective indices of the fundamental core mode in the MTIR guidance regime nMTIR(crosses on Fig. 3) as well as inside the 1st bandgap nBG(triangles) are also plotted. The material dispersion has not been considered for these calculations. The air holes radius r is fixed to 7.58×10-2Λ, corresponding to a value close to the one obtained during the stacking step of the fabrication process. This air holes size is large enough to obtain a spectral region for which nFSM (broken lines) is below the core refractive index equal to 1.45 (λ/Λ > 0.47), enabling MTIR guidance in this domain (dark gray region in Fig. 3). Note however that a MTIR guided mode arises only if the fiber core is sufficiently large[20] which is the case of our structure, the core corresponding to a defect in the periodicity where both a high index region and the first 6 air holes are removed. As can be seen, the effective index of the MTIR guided mode nMTIR (crosses on Fig. 3) stays relatively close to 1.45 and has a low dispersion while the effective index nBG of the fundamental mode inside the first bandgap (triangles) is significantly lower than nMTIR on most of the bandgap width and has also a higher dispersion.
Fig. 3. Schematic transversal cut and associated band diagram for the hybrid guiding PCF. The three first bandgaps are shaded in light gray. Also shown the fundamental space filling mode index (dashed line). The region where MTIR guidance is possible is shaded in dark gray. The effective index of the fundamental core mode in the 1st bandgap (triangle) and the fundamental MTIR guided core mode (cross) are also plotted. The material dispersion has not been considered here.
The effective index difference (Δnwaveguide) between these two modes is shown on Fig. 4, for SHG and for THG. Δnwaveguide proves to have the right sign and to be large enough to counterbalance the material dispersion (see Fig. 1). More precisely the colored part of both curves on Fig. 4 indicates the regions where the index difference is sufficient to compensate the material dispersion (contrary to the gray part). For increasing values of λ/Λ in Fig. 4, both MTIR guided and bandgap guided modes are obviously pushed to longer wavelengths. The maximal value of Δnwaveguide is then reached when nBG is closed to the upper bandgap border.
4. Basic properties in the case of few designs
To illustrate the possibility offered by our fiber design, we discuss below three fiber structures satisfying the phase index matching condition for SHG in the case of the most common laser sources, λ=0.8, 1.06 and 1.5 µm. Note that these structures are obtained simply by adjusting the fiber pitch, i.e. all the structures are homothetic. The main linear characteristic values for each design are summarized in Table 1. In our case, the phase velocity mismatch being zero (infinite coherence length), one has to consider the group velocity mismatch, Δv G=|vG(λfund)-vG(λharm)| with vG(λ)=c/(neff(λ)-λ∂neff/∂λ) (characterized by the temporal walk-off length LW=Tο/Δv G, with Tο, the temporal width for the pulse[17]) and the chromatic group velocity dispersion of the two waves (characterized by the dispersion length LD[17]) as these linear effects can limit the useful fiber length. The temporal width for the pulse T0 has been fixed to 1 ns. In that case, the dispersion lengths LD are not a limiting factor as they exceed hundreds of kilometers (not shown in Table 1). The values of L W, obtained for T0=1 ns, are between 3.6 and 10.8 m, allowing us to use fiber lengths in the meter range. Eleven rings of high index rods are required in order to obtain confinement losses below 1 dB/m for the fundamental wave at 1.06 and 1.5 µm (the losses at the harmonic wave being very low, respectively 0.52 dB/km and 0.47 dB/km). Note however that, by adding an extra air holes ring in the cladding[22], one may decrease the number of high index rings while reducing significantly both the confinement and bend loss of the fibre. We can also notice that the confinement losses for both λ fund and λ harm are significantly higher in the case of λ fund=0.8 µm. This is due to the fact that the index mismatch to compensate is higher at this short wavelength (Fig. 1). Indeed higher Δnmaterial imposes to work with a harmonic mode deeper in the first bandgap (and so closer to the bandgap edge: see Fig. 3 and 4) and also with a fundamental mode closer to nFSM. As a consequence, the confinement losses of both modes substantially increase for short fundamental wavelengths.
Fig. 4. Index difference Δnwaveguide in the hybrid guiding PCF between a fundamental wavelength λ fund, guided by the total reflection effect, and its second or third harmonic λ harm, guided inside the first bandgap. The colored part of both curves indicates that Δnwaveguide is sufficient to compensate Δnmaterial·Δnwaveguide=n(λ fund)-n(λ harm).
Table 1. Designs fulfiling the phase index matching condition for SHG.
Figure 5 represents the intensity profiles of the fundamental MTIR guided mode at 1500 nm (red line) and its second harmonic bandgap guided mode at 750 nm (green line) along the two principal directions (x1, x2) defined on the fiber cut of Fig. 3. The MTIR guided fundamental mode exhibits an intensity profile which is larger than that of the harmonic one. This result was predictable as the MTIR guided mode required the presence of air holes while the bandgap guided mode is mainly guided by the presence of the high index rods: the first rods being closer to the central defect that the first air holes, one should expect the mode field diameter of the bandgap guided mode to be smaller than the MTIR guided mode. The typical resonances of a bandgap guided mode clearly appear in the index rods of the first ring (see harmonic profile along x1). However, in the silica core region of the fiber, where most of the intensity is concentrated, the harmonic mode exhibits a quasi-gaussian intensity profile[9], making it directly suitable for many applications. This is the first time, to our best knowledge, that the phase index matching condition between two fundamental modes is perfectly fulfiled inside an optical fiber.
Fig. 5. Normalized intensity profiles along the two spatial cross sections defined in Fig. 3, for the fundamental mode at 1500 nm (red line) and its phase index matched second harmonic at 750 nm (green line).
Fig. 6. Tunability of the phase index matching condition for SHG and THG: evolution of Λ as a function of the fundamental wavelength λ fund for various ratio r/Λ.
5. Tunability of the phase index matching condition
Lets finish our study by looking at the tunability of the phase index matching condition by modifying the fiber pitch. The solid curves on Fig. 6 show the evolution of the pitch required to satisfy the phase index matching for SHG or THG versus its corresponding fundamental wavelengths λ fund. Note that the phase matching can be satisfied for λ fund > 3 µm and λ harm < 0.3 µm, but because of the silica transparency window we decide, somehow arbitrarily, to limit the plot to fundamental wavelengths between 0.6 and 2 µm. As can be seen, the phase matching condition is tunable over a very large spectral range, and this by modifying only the dimension of the structure. Moreover, when the air holes size slightly changes (dashed and dotted curves), the tunability is retained, which is satisfying as an accurate control of the ratio r/Λ during the fabrication is very challenging. Note that if the fabrication of this kind of fiber, with very small air holes (about 185 nm for SHG and λ fund=1.06 µm) is expected to be difficult, it should still however be feasible as for example G. S. Wiederhecker et al. present results on few meters long fiber with a central hole diameter as small as 110 nm[23]. We can also notice from Fig. 6 that, for a fixed λ fund, Λ decreases when the ratio r/Λ increases. This is due to a decrease of nMTIR when r/Λ increases, while the bandgap mode is significantly less affected, reducing thus the waveguide index difference. It is then necessary to decrease Λ to compensate this diminution of Δnwaveguide (Fig. 4).
6. Conclusion
A hybrid guiding PCF that allows to satisfy the phase index matching condition for SHG or THG has been modelled. The main advantage of this new design is that the phase matching occurs between two fundamental modes and the generated mode exhibits a suitable gaussian-like intensity profile. This result is achieved thanks to the coexistence of two guidance mechanisms. Indeed, while the harmonic wave is confined by bandgap effect, the fundamental wave is guided thanks to the total reflection mechanism. Practical fiber designs for SHG have been proposed and their basic linear characteristics given. Moreover, we show that the phase matching condition is tunable over a very large spectral range simply by tuning the pitch. Such a structure proves to be particularly attractive for harmonic generation.
Acknowledgments
This work was supported by the “Conseil Régional Nord-Pas de Calais”, the “Fonds Européen de Développement Economique des Régions”, the “Agence National de la Recherche” (ANR-05-BLAN-0080).
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