We demonstrate continuous-wave wavelength conversion through four-wave mixing in an endlessly single mode photonic crystal fiber. Phasematching is possible at vanishing pump power in the anomalous dispersion regime between the two zero-dispersion wavelengths. By mixing appropriate pump and idler sources, signals in the range 500–650 nm are obtained in good accordance with calculated phasematching curves. The conversion efficiency from idler to signal power is currently limited to 0.3% by the low spectral density of the pump and idler sources at hand, but can be greatly enhanced by applying narrow line width lasers.
©2004 Optical Society of America
CorrectionsThomas V. Andersen, Karen Marie Hilligsøe, Carsten K. Nielsen, Jan Thøgersen, K. P. Hansen, Søren R. Keiding, and Jakob J. Larsen, "Continuous-wave wavelength conversion in a photonic crystal fiber with two zero-dispersion wavelengths: erratum," Opt. Express 13, 3581-3582 (2005)
Photonic Crystal Fibers  (PCFs) have received a lot of attention in the last few years, partly due to their interesting nonlinear properties. The demonstration of supercontinuum generation (SCG) with pulses from Ti:Sapphire femtosecond oscillators [2, 3] has enabled the realization of new fiber based light sources. While SCG in PCFs is now well understood [4, 5, 6, 7, 8, 9, 10] and has found use in metrology , CARS microscopy  and tomography , the huge potential of these fibers remains largely unexploited. The small core size and long interaction length of the fibers result in a high nonlinear response which allows us to study and use otherwise weak nonlinear effects. The outcome may be new types of nonlinear devices such as wavelength converters and optical switches .
Four-wave mixing (FWM) is a well known nonlinear process in which Stokes and anti-Stokes photons are generated from two pump photons through the third order susceptibility. The conversion efficiency of the process depends crucially on a phasematching condition, which in turn depends on the dispersion properties of the fiber. When phasematched, FWM has higher gain than stimulated Raman scattering (SRS) and can be used for upconversion as well as downconversion . For that reason, FWM has been exploited to make fiber optic parametric oscillators (FOPOs), which can generate wavelengths otherwise not attainable with conventional fiber lasers.
Already in 1987 Margulis obtained lasing through FWM by placing an optical fiber in an external cavity and pumping it with 100 ps pulses . More recently, Sharping  has reported a FOPO based on FWM with 600 fs pulses in a PCF. In 2002 Marhic  demonstrated a FOPO using continuous wave FWM. By writing fiber Bragg gratings (FBG) into a 100 m long piece of a highly nonlinear fiber, they achieved parametric oscillation with only 240 mW of pump power at 1563 nm. The FBGs made a cavity for the signal wavelength at 1560 nm, and the idler, being the useful output, could be tuned 80 nm by varying the pump wavelength.
Very recently Matos  has demonstrated an all-fiber-integrated OPO through continuous wave FWM in a PCF. A cavity was built around a 100 meter long PCF and oscillation was achieved within 2 nm from the pump at 1552 nm. The threshold pump power was 1.28 W.
In this paper we present continuous wave FWM in a polarization maintaining PCF with two zero-dispersion wavelengths (ZDW). The dispersion profile has profound influence on the nonlinear processes taking place in the fiber as described in Ref. . Specifically, it turns out that phasematching for FWM is obtained in the entire positive dispersion region between the two ZDWs. A tunable output in the range 500 nm–1800 nm is in principle possible by pumping the fiber between the ZDWs while seeding with an appropriate idler. The large span of phasematched wavelengths may prove useful for broadband amplification and we therefore suggest to implement this fiber in a FOPO as well.
2. Four-wave mixing
where Ap, Ai and As are the field amplitudes of the pump, idler and signal respectively, and γ is the nonlinear coefficient, γ=n 2 ω/A eff c. n 2 is the nonlinear index of refraction, ω is the angular frequency of the wave (pump, signal or idler) and A eff is the effective mode area. α is the loss coefficient according to P(z)=P(0) exp(-αz), and Δβ is the difference in the propagation constants given by Δβ=βs+βi-2βp. Equations (1) and (2) generally have to be solved numerically but assuming un-depleted pump it is possible to obtain an analytical expression for the signal power [15, 21]
where g is the gain:
and κ is given by
where P 0 is the pump power. Maximum gain of g=γP 0 is obtained when κ=0, which occurs when the nonlinear phase shift 2γP 0 is compensated by a negative wave vector mismatch, Δβ. Expanding β(ω) around the pump frequency results in the phasematching condition:
where β 2, β 4 and β 6 are the second, fourth and sixth derivative of β with respect to ω evaluated at the pump frequency and Ωs is the frequency shift from the pump to the Stokes/anti-Stokes frequencies.
When analyzing conventional single mode fibers, it is customary to include only the second derivative of β in Eq. (6), since the higher order terms are generally small. In this approximation, phasematching can only occur in the anomalous dispersion regime where a negative β 2 can compensate for the positive nonlinear phase shift . In dispersion-shifted PCFs the FWM scheme can be slightly more complicated. The unusual dispersion profiles imply that the higher order terms in Eq. (6) can not be neglected . Depending on the signs of β 2, β 4, β 6 etc., exact phasematching is no longer restricted to the anomalous dispersion regime, but can also be found several nanometers below the ZDW - even in absence of a nonlinear phase contribution [8, 22].
In this paper we present work on a PCF with two ZDWs at 755 nm and 1235 nm respectively. The fiber is made of pure silica using the stack and draw technique. Figure 1 (right) shows the cross section of the fiber. The core has a diameter of 1.8µm and is surrounded by six air holes of which two are larger than the rest. This causes strong birefringence of Δn>3·10-4 and the fiber is therefore polarization maintaining. The pitch of the fiber is 1.2 µm while the average hole size is 0.55µm. The nonlinear coefficient, γ is 95 (Wkm)-1 (@780 nm) . The strong curvature of the dispersion profile (Fig. 1 (left)) implies that the higher derivatives of β (ω) are important. Based on a fitted Taylor expansion, integration of the dispersion allows us to solve the phasematching equation (Eq. (6)). In this particular case, a better polynomial fit can be made to β 2(λ) than to β 2(ω). The polynomial fit to the data shown in Fig. 1 is given by
be calculated. Figure 2 (left) shows the calculated β 2, β 4, β 6 and β 8 as functions of λ in the wavelength interval between the two ZDWs. The shown curves have been scaled according to
in order to show the relevance of each term in Eq. (6). The figure indicates how the significant size, and the changing signs of the higher order terms, may result in phasematching in the anomalous dispersion regime even without a nonlinear phase contribution. Figure 2 (right) indicates that phasematching may also take place in the normal dispersion regime close to the ZDW where the negative β 4 and β 8 can compensate for the positive β 2 and β 6.
Figure 3 (left) shows a phasematching diagram based on the calculations above. Phase-matched signal and idler wavelengths are found on vertical lines for a given pump wavelength. The nonlinear contribution is neglected in this figure (P 0=0). The curve is seen to be quite different from phasematching curves for PCFs with only one ZDW [8, 22], for which phase-matching at low power is only possible slightly below the ZDW . In the present fiber, however, phasematching takes place in the entire range of anomalous dispersion between the two ZDWs and several nanometers into the high-frequency normal dispersion region. At higher pump powers the phasematching is eventually perturbed by the nonlinear contribution 2γP 0. Figure 3 (right), shows the resulting phasematching curve when the fiber is pumped by 1000W. A new set of closely lying phasematched wavelengths appears, while the wavelengths already matched at zero pump power, remain matched. The range of pump wavelengths, which can be phasematched is seen to span more than 450 nm, and the phasematched wavelengths are found between 500 nm and 1800 nm. However, in the present fiber confinement losses appear over 1600 nm, so for long lengths of fiber the useful wavelength interval narrows somewhat. Through proper design of the microstructure of the fiber, such losses can be greatly reduced to enable long-wavelength operation .
It should be noted that the dispersion is only measured from 600 nm to 1550 nm so any prediction outside this range depends on the accuracy of the extrapolations made. Furthermore, the dispersion is measured by white light interferometry [25, 26] using an unpolarized light source. The curve in Fig. 1 is therefore an average of the dispersion in the two polarization axes of the fiber (arbitrarily labelled as major and minor axis respectively). Close to the low-wavelength ZDW at ~755 nm the phasematching curve is particularly interesting since it predicts that phasematched wavelengths can be generated continuously between 500 nm and 1500 nm by tuning the pump laser 15 nm - see Fig. 4. Furthermore, there are pump wavelengths for which two sets of phasematched wavelengths are possible. This multiple frequency generation has previously been predicted [27, 28, 29], but has, to our knowledge, not yet been found experimentally. Figure 3 also indicates that multiple frequency generation is possible in the entire negative dispersion region when the fiber is pumped with high power. It would be interesting to study this phenomenon in more detail since it could bring further understanding of the SCG processes in this class of fiber .
4. Experimental results
To demonstrate the speculations above, outputs from a Ti:Sapphire laser operating at 750–830 nm and a laser diode centered at 975 nm were overlapped and coupled into a 20 meter long piece of the PCF with a microscope objective (see Fig. 5). The Ti:Sapphire laser is a MIRA 900 from Coherent which is designed for pulsed operation, but can be forced to operate in CW mode. However, the design of the cavity causes a broad line width of 4 nm and the resulting low spectral density is not ideal for nonlinear experiments. To ensure excitation of only one polarization mode, the fiber was mounted in a rotational stage and aligned with its major axis parallel to the polarization of the Ti:Sapphire laser. Light from the 975 nm diode was polarized in the same orientation by appropriately bending the delivery fiber . After the PCF, the beam was collimated and dispersed by a prism allowing power measurements of the individual colors.
Figure 6(a) shows the experimentally obtained spectrum at phase-matching along the major axis of the fiber. The matched wavelengths are 774 nm (pump), 975 nm (idler) and 641 nm (signal). Similar experiments were made with idler sources at 1064 nm, 1312 nm, 1493 nm and 1549 nm, and the resulting spectra can be seen in Figs. 6(b)–(e). All signal outputs were found to be linearly polarized along the same axis as the pump light leaving the fiber, meaning that the birefringence of the fiber keeps the light in one polarization state. Coupling into the minor axis of the fiber resulted in a similar group of phasematched wavelengths and the outputs were once again found to be linearly polarized parallel to the pump. Figure 7 shows the measured sets of phasematched wavelengths along the major and minor axis respectively. The filled squares indicate measured values while the dotted line is the theoretical curve previously shown in Fig. 4. As mentioned before, the theoretical curve is based on the average dispersion in the two axes and does therefore not relate directly to any of the measured sets of phasematched wavelengths. However, the qualitative shape of the phasematching curve is readily recognized in both axes (dashed lines) and is thus useful in predicting where phasematching can be found. The data also indicate that it should be possible to obtain phasematching for two sets of wavelengths at the same time (as illustrated in Fig. 4).
To obtain efficient conversion from idler to signal power, it is important to match the line widths of the idler and pump sources involved. Figure 8 (left) shows a close-up of the phase-matching curve in the region of 1493 nm. Due to the steep slope of the curve, modes in the 10 nm wide spectrum of the laser diode can be phasematched to pump modes within a 1 nm interval of the pump spectrum. Since the pump has a spectral width of 4 nm, only a part of the pump power actually participates in the mixing process which lowers the conversion efficiency. Figure 8 (right) shows the output power at 518 nm as a function of pump power, when phasematching with 769 nm and 1493 nm is obtained along the minor axis of the fiber. The curve is in qualitatively accordance with analytical approximations presented in , and shows that the signal is indeed generated through a FWM process. A maximum of 12µW at 518 nm was achieved when the fiber was pumped with 200 mW at 769 nm and seeded with 40 mW at 1493 nm. This corresponds to a conversion efficiency from idler to signal power of 0.3%, which is lower than expected from numerical integration of Eqs. (1) and (2) where the spectral width of the sources has been neglected. It should be noted that fluctuations in the structure of the fiber may cause variations in the phasematching condition along the fiber. This is known to cause a reduction in bandwidth and gain of the FWM process .
5. Discussion and conclusion
It is clear that a spectrally narrow pump source would greatly enhance the conversion efficiency found above. A “single frequency” laser would make most of the pump power available for the mixing process, and numerical solutions of Eqs. (1) and (2) show that 200 mW of pump power, combined with 40 mW of idler power, would result in 2 mW at the signal wavelength. This corresponds to a conversion efficiency of 5% which would be quite acceptable for many purposes. With the advent of high power, narrow line width diode lasers  it will be possible to obtain such conversions with simple and compact setups. An optical parametric oscillator (OPO) as described in Ref. [17, 19] could be made to obtain tunable and efficient continuous-wave wavelength conversion. This may give access to otherwise unattainable parts of the spectrum  and be useful in spectroscopy, quantum optics and telecommunication.
Emphasizing the potential of this specific fiber, it should also be noted that the ability to control the dispersion properties of the fibers, makes it possible to design new fibers with other phasematching properties. The freedom to move the ZDWs can be used to make the zones of high tunability coincide with available pump sources. Furthermore, improved understanding of the correlation between dispersion profiles and phasematching curves, will give more design freedom and may enable access to the blue part of the spectrum.
In conclusion we have demonstrated continuous wave FWM in a PCF with two zero-dispersion wavelengths. The dispersion profile of the fiber allows phasematched degenerate FWM to take place in the anomalous dispersion regime at low CW pump powers. In the present experiments, phasematching at five pump wavelengths was demonstrated, which indicates that a large span of wavelengths can be matched by tuning the pump laser only a few nanometers in the vicinity of the lowest ZDW. Simultaneous phasematching of two sets of wavelengths also seems feasible but cannot be verified with the equipment at hand. A conversion efficiency of 0.3% from an idler at 1493 nm to a signal at 518 nm has been demonstrated but is severely limited by the low spectral density of the pump. Appropriate choice of pump and idler sources will greatly enhance performance and allow efficient wavelength conversion.
T. V. Andersen, K. M. Hilligsøe and C. K. Nielsen would like to thank NKT Academy for financial support.
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