Intermodal four-wave mixing (FWM) in microstructured optical fibers (MOF) is studied theoretically and experimentally. The dependance of FWM frequency detuning on the geometrical parameters of the fiber, namely the pitch, the core width and the air-filling fraction is derived. We propose to use the results of this investigation to control the position of the Stokes and anti-Stokes waves directly from the fiber transverse structure drawing without the need for time-consuming simulations as in usual design procedures. Stokes sideband can then be freely tuned within the S-, L-, and C- bands with great potential for infrared applications.
©2008 Optical Society of America
The microstructured optical fibers (MOFs) offer huge advantages over the conventional step-index fibers for applications based on four-wave mixing (FWM) [1–6], such as frequency converters , parametric oscillators , supercontinuum sources , and quantum information devices . Indeed, the core size in MOFs is dramatically reduced, that results in increasing FWM efficiency by at least two orders of magnitude. But the most unique advantage of MOFs is the possibility of tailoring the dispersion curves by simply tuning air holes size and pitch [11–14], that in turn determines the position of the FWM sidebands [15,16]. Therefore, the fiber transverse structure has to be carefully designed to match the requirement of phase-matching at the wavelength of operation for the given application. However, usual design procedures remain empirical. In practice, one has to perform a large number of numerical simulations from different fibre transverse structures, so as to calculate the associated dispersion curves and subsequently, the phase-matched wavelengths. In more details, starting from an arbitrary fibre transverse structure, one has to increase or decrease air holes pitch and diameter until the targeted operation wavelength is phase-matched [17,18]. Unfortunately this procedure is time-consuming, since a large number of simulations is needed, with no guarantee to end up the design procedure with a realistic fiber structure.
We demonstrate that Stokes and anti-Stokes wave positions arising from a unique large band modal FWM process can be directly and easily deduced from the principal fiber parameters by using specific abacus. Our experimental and numerical investigation is based on a set of MOFs having slightly different transverse structures, in which we have investigated intermodal FWM . We analyse in details the influence of the fiber geometrical parameters on phase-matched wavelengths. We propose to use the results of this study to control the position of the Stokes and anti-Stokes sidebands directly from the fiber profile, in order to avoid lengthy numerical calculations of the dispersion curves. For example, a simple evaluation of the core diameter for a given set of the air hole diameter and pitch enables to predict the sidebands positions of modal FWM process. Hence, by pumping at 1064 nm, we observe that the anti-Stokes sideband can be freely tuned in the range of wavelengths between 650 nm and 850 nm, depending on the fiber geometrical parameters. The corresponding Stokes sideband can be observed between 1400 nm and 1750 nm, covering the whole S- (short), C- (central), and L- (long) telecom bands.
2. Experimental study
The different air-silica MOFs, hereafter referred as “F1” to “F13”, have been manufactured by using the stack and draw technique. Their geomatrical parameters are given in Tab. 1. Two enlarged air holes have been pinpointed in the first ring of air holes to induce strong birefringence, as depicted in Fig. 1.
The symbol Λ denotes the distance between two adjacent cladding holes (Fig. 1). The air hole diameter and the core width are denoted by d and Γ, respectively. Our MOFs have been drawn with different values of Γ, Λ, and d. To reduce or increase the core width, one can enlarge or shrink the two birefringence holes, respectively. In Fig. 1, the principal axes of birefringence have been denoted by x and y for the fast and slow axes, respectively.
In this experiment, we use a Q-switched Nd:YAG microchip laser (Teem Photonics NP-10620) for pumping several 80 cm-long fiber samples at 1064 nm. The linearly polarized laser pump is equally divided into fundamental LP01 and high-order LP11 modes through a selective coupling at the input end of the MOF. The corresponding Stokes and anti-Stokes sidebands are generated in LP01 and LP11 modes, respectively, in contrast with Ref. . Such modal distribution is due to the fiber dispersion, which falls in the anomalous regime at the pump wavelength for each fiber sample. Figure 2 shows the theoretical dispersion curve of LP 01 and LP11 modes polarized along the x-axis for F13 fiber sample and the corresponding phase matching diagram. Note the absence of bifurcation at the zero-dispersionwavelength in contrast to what is usually observed in the conventional single-mode FWM process.
In Fig. 3, we show a selection of examples of the output spectra obtained for different MOFs. The output spectra of fibers F13, F12, F3, and F9, are represented by curves (a), (c), (d), and (e), respectively, and those of fiber F6 are shown by curves (b) and (f). In Fig. 3(a), (b), (c), (d), and (e) the laser pump was entirely polarized along the fast axis. In Fig 3(f), the laser pump was entirely polarized along the slow axis. From Fig. 3, we see that the anti-Stokes sidebands can appear in the range 650–850 nm while the corresponding Stokes sidebands vary between 1400 nm and 1750 nm. This ranges the whole S-, C-, and L- bands. The nonlinear conversion recorded for all 80 cm-long fiber samples is measured between 5% and 10%.
3. Discussion and design issues
In Fig. 4, we report with circle and square marks the different FWM frequency detuning from the laser pump using different MOFs. In the later, the pump wave is always polarized along the fast axis of the MOFs. We have also represented the evolution of the FWM frequency detuning (solid and dashed lines) calculated from the phase-matching relation using the effective index curves for the modes that were determined with a finite-element based mode solver. The related error bars in Fig. 4, give the uncertainty on the measurement of the MOFs parameters arising from the SEM pictures. The small errors on the measurement of the FWM frequency detunings coming from the optical spectrum analyzer are here neglected. In Fig. 4(a), we represent the evolution of the FWM frequency detuning as a function of the fiber core width Γ. The circles indicates the measured FWM frequency detunings for fibers F1, F7, F8, F9, F10, and F11, i.e. Λ⋍2.5µm and d⋍0.67Λ. Our experimental results agree with the theory of phase-matching, here represented by a solid curve. Figure 4(a) indicates that the FWM frequency detuning scales almost linearly with the core width Γ. Indeed, the frequency detuning grows from 70 THz to 120 THz while the core width Γ increases from 1.4µm to 2.1µm. It is clear from this result that FWM frequency detuning can be strongly influenced by the choice of the core width Γ.
To discuss the effect of pitch size, we show (as squares) the measured frequency detunings for fibers F5 and F6, i.e. Λ⋍3.63µm and d⋍0.67Λ. Dashed line represents the corresponding calculated FWM frequency detuning. The comparison between solid and dashed lines reveals that a 44 % increase in the pitch value results in a 45 THz decrease in the frequency detuning.
Figure 4(b) illustrates the detailed evolution of the frequency detuning as a function of the pitch Λ. The circles indicate the measured frequency detunings for fibers F2, F3, F4, F5 and F6, i.e. Γ⋍2.3µm and d⋍0.66Λ. The theory which is represented as a solid line is in good agreement with our experiment. In practice by increasing the pitch by 43%, one can get a 40 THz decrease in the frequency detuning, in agreement with Fig. 4(a). In Fig. 4(b), we have also indicated the measured frequency detunings for fibers F7, F8 and F9, i.e. Γ⋍1.88µm and d⋍0.68Λ, which are represented by squares. The corresponding theoretical values are represented by the dashed line. The comparison between solid and dashed curves confirms that the FWM frequency detuning grows larger with the fiber core Γ. However, the frequency detuning can be kept nearly constant, whether a reduction of the core size Γ is compensated by a corresponding reduction in pitch size Λ.
Finally, we show the evolution of the frequency detuning as a function of the air-filling fraction in Fig. 4(c), where the circles indicate the measured values for fibers F1, F10 and F12, i.e. Γ⋍1.58µm and Λ⋍2.6µm. Solid curve represents the calculated frequency detuning. We observe that a growth of the air-filling fraction by 30 % produces an increase of 25 THz in the FWM frequency detuning.
The presence of two enlarged air holes near the core has a dramatical influence on the waveguide dispersion. Therefore, the core width Γ, which is increased by simply reducing the diameter of these two air holes, has a dominant influence on the FWM frequency detuning. By contrast, the pitch or the air-filling fraction in the cladding are less critical parameters.
In conclusion, we have investigated intermodal FWM in different samples of MOFs. We have demonstrated, experimentally and theoretically, how FWM sideband position can be easily controlled by varying only one of the MOFs parameters like pitch, core diameter or air-filling fraction. The trends shown in this study are applicable to π/3 symmetric profile which does not present two larger holes. Nevertheless, in this kind of fiber, it becomes difficult to induce a unique and efficient modal FWM. Our extensive study provides useful information on how to design the MOFs for further applications in the infrared. Indeed, the numerical results summarized in Fig. 4 can further be exploited as a convenient and fast tool to determine the suitable fiber profile for any targeted wavelength in the S-, L- or C-band.
The authors acknowledge the financial support from the Conseil Régional du Limousin and from the Information Society Technologies (IST) priority of the European Commission’s 6 th Framework Program within the frame of NextGenPCF integrated project.
References and links
1. A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum. Electron. 16, 694–697 (1980). [CrossRef]
3. J. E. Sharping, M. Fiorentino, P. Kumar, and R. S. Windeler, “Optical parametric oscillator based four-wave mixing in microstructure fiber,” Opt. Lett. 27, 1675–1677 (2002). [CrossRef]
4. J. E. Sharping, M. Fiorentino, A. Coker, P. Kumar, and R. S. Windeler, “Four-wave mixing in microstructure fiber,” Opt. Lett. 26, 1048–1050 (2001). [CrossRef]
5. G. Millot, A. Sauter, J. M. Dudley, L. Provino, and R. S. Windeler, “Polarization mode dispersion and vectorial modulational instability in airsilica microstructure fiber,” Opt. Lett. 27, 695–697 (2002). [CrossRef]
6. S. Coen, A. H. L. Chau, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation by stimulated Raman scattering and parametric four-wave mixing in photonic crystal fibers,” J. Opt. Soc. Am. B 19, 753–763 (2002). [CrossRef]
7. K. K. Chow, K. Kikuchi, T. Nagashima, T. Hasegawa, S. Ohara, and N. Sugimoto, “Four-wave mixing based widely tunable wavelength conversion using 1-m dispersion-shifted bismuth-oxide photonic crystal fiber,” Opt. Express 15, 15418–15423 (2007). [CrossRef] [PubMed]
9. C. Lesvigne, V. Couderc, A. Tonello, P. Leproux, A. Barthélémy, S. Lacroix, F. Druon, P. Blandin, M. Hanna, and P. Georges, “Visible supercontinuum generation controlled by intermodal four-wave mixing in microstructured fiber,” Opt. Lett. 32, 2173–2175 (2007). [CrossRef] [PubMed]
11. G. K. Wong, A. Y. Chen, S. Ha, R. Kruhlak, S. Murdoch, R. Leonhardt, J. Harvey, and N. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express 13, 8662–8670 (2005). [CrossRef] [PubMed]
12. N. G. R. Broderick, R. M. Monro, P. J. Bennett, and D. J. Richardson, “Nonlinearity in holey optical fibers: measurement and future opportunities,” Opt. Lett. 24, 1395–1397 (1999). [CrossRef]
13. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Scalar modulational instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. 28, 2225–2227 (2003). [CrossRef] [PubMed]
15. G. P. Agrawal, Nonlinear fiber optics2nd Ed. (Academic, Boston, Mass. 1995).
16. R. H. Stolen, “Phase-matched-stimulated four-photon mixing in Silica-fiber waveguides,” IEEE J. Quantum Electron. 11, 100–103 (1975). [CrossRef]
17. N. I. Nikolov, T. Srensen, O. Bang, and A. Bjarklev, “Improving efficiency of supercontinuum generation in photonic crystal fibers by direct degenerate four-wave mixing,” J. Opt. Soc. Am. B 20, 2329–2337 (2003). [CrossRef]
18. M. H. Frosz, T. Srensen, and O. Bang, “Nano-engineering of a photonic crystal fiber for supercontinuum spectral shaping,” J. Opt. Soc. Am. B 223, 1692–1699 (2006). [CrossRef]