## Abstract

In this paper the length of a photonic crystal fiber is optimised to perform high average output power parametric generation with maximum efficiency. It is shown that the fiber length has to be increased up to 150 m, well beyond the walk-off distance between the pump and signal/idler, to optimize the generation efficiency. In this regime, the Raman process can take over from four-wave mixing and lead to supercontinuum generation. It is shown that the parametric wavelength conversion is directional; probably due to small variations in the core dimensions along the fiber length. The fiber exhibits up to 40% conversion efficiency, with the idler (0.9 µm) and the signal (1.3 µm) having a combined output power of over 1.5 W.

© 2008 Optical Society of America

## 1. Introduction

Non-linear frequency conversion in standard optical fibers using the Four Wave Mixing (FWM) process was first observed some time ago [1]. The pump wavelength has to be in the vicinity of the Zero Group Velocity Dispersion (ZGVD) wavelength of the fiber mode to achieve the phase matching conditions. Using Photonic Crystal Fiber (PCF) technology, it is possible to tune the ZGVD wavelength from 0.5 µm (using submicron diameter fibers [2]) to 1.3 µm, allowing greater flexibility for the pump wavelength.

Since its first observation [3], FWM in PCFs has been optimised in different ways, including degenerate FWM [4], non-degenerate FWM [5] and polarisation dependent FWM in birefringent fibers [6]. Recently an efficient source of correlated [7] and entangled [8] photon pairs has been demonstrated using FWM in PCF. Using long fibers, peak power thresholds of less than 1 W for the FWM process are achievable and efficient non-linear frequency conversion around 1.5 µm has been achieved in Optical Parametric Oscillators (OPO) by pumping CW [9]. The newly generated wavelengths can be tuned over a very large range by tuning the pump wavelength by a few tens of nanometers [10]. Efficient tunable OPOs have been demonstrated recently [11, 12].

In this paper, we focus our attention on the optimisation of parametric generation in a PCF to produce high average output power. Of the various pulse formats possible, femtosecond pulses produce very high peak powers but the spectral width can extend several nanometres. As a consequence, the FWM bands have large bandwidths. At the other end of the scale, pulses in the nanosecond range have a very narrow linewidth and are long enough to ignore walk-off issues [4]. But the repetition rate of such sources is only a few kHz and the pulse energy is large enough to damage the fiber end. It is therefore difficult to reach average powers in the watt range using nanosecond pulses. Pulses having durations of a few tens of picoseconds from high repetition rate mode-locked lasers seem to be the best pulse format for achieving nonlinear frequency conversion with high average powers in PCFs [13]. Here we study the optimisation of parametric generation in a PCF pumped by a mode-locked Nd:YVO4 laser emitting 30 ps long pulses at a repetition rate of 250 MHz and delivering up to 7 W of average power at 1064 µm.

## 2. Fiber dispersion and experimental set-up

A nonlinear PCF originally 150 m long was used for these experiments. The fiber, fabricated by Crystal Fiber A/S, is shown in Fig. 1. The measured pitch of the photonic crystal structure is 2.47 µm and the hole diameter is 0.90 µm. The resulting core diameter is 4.1 µm.

Figure 2(a) shows that the losses of the fiber are about 20 dB/km between 0.8 µm and 1.6 µm apart from the strong OH absorption at 1.4 µm. This fiber has been specifically chosen for its measured dispersion properties (see Fig. 2(b)). Its zero group velocity dispersion is at 1070 nm, which is slightly higher than the 1064 nm pump source we intend to use.

The gain for degenerate FWM in a fiber is provided by the third order nonlinearity; that is the nonlinear refractive index of silica, *n*
_{2}. The phase matching conditions for degenerate FWM are:

where *β* is the propagation constant in the fiber, *ω* is the radiation frequency, *P* is the optical power and *γ* is the nonlinear coefficient of the fiber defined by:

where *n _{2}* is the nonlinear refractive index,

*λ*is the pump wavelength and

*A*is the effective mode field area of the guided mode.

_{eff}We have modeled the dispersion properties of the PCF using the measured characteristics (pitch and hole diameter). A scalar field approach is used to calculate the effective index of the PCF cladding [14] as a function of wavelength. Next, the resulting step index waveguide problem is solved in standard fashion [15]. This study reveals that the mode field diameter of the propagating mode at 1.064 µm is 7 µm (determined at the 1/e^{2} point). The evolution of the propagation constant *β* as a function of wavelength was calculated, from which we deduced the dispersion parameter *D*. The theoretical wavelength dependence of *D* was found to closely match the experimental measurement shown in Fig. (2b). In particular, the modeling gives a value of 1071.5 nm for the ZGVD wavelength, very close to the measured value of 1070 nm. The difference can be attributed to the uncertainty in the measurement of the pitch of the structure. Tuning the pitch ever so slightly gave perfect agreement with the measured ZGVD.

The calculated curve for *β*(*λ*) was used to solve Eqs. (1) and (2). The resulting phase matching diagram calculated for peak powers of 0 W and 500 W is shown in Fig. 3. The new spectral components generated when the pump wavelength is lower than the ZGVD wavelength are associated with the FWM process.

A mode-locked Nd:YVO4 laser emitting up to 7 W of average power at 1064 nm at a repetition rate of 250 MHz was used to pump the PCF. The pulse duration of the laser has been measured using a commercial autocorrelator (Femtochrome). The pulses are Fourier transform limited with a pulse duration of 27 ps at Full Width Half Maximum (FWHM) and the corresponding bandwidth of the laser is 0.04 nm. As a consequence the maximum peak power available from the laser is 1000 W. The laser was coupled into the photonic crystal fiber using a 0.65 NA microscope objective with a typical coupling efficiency of 50%. As a result, the maximum peak power inside the fiber was 500 W. The power coupled into the fiber was controlled using a waveplate and polariser. The output of the fiber was coupled into an optical spectrum analyser (Anrisu model MS9701B for the results of section 3 and Ando Model AO6317C for the results of sections 4, 5, and 6), whose resolution was set at 0.02 nm. The detection range of the OSAs spanned from 0.6 µm to 1.7 µm.

## 3. FWM using a 10 m long fiber sample

When high peak powers of tens of kW are available, short fibers can be used for FWM or supercontinuum generation. A standard Ti:sapphire laser provides up 100 kW peak power in a 100 fs long pulse and a few cm of nonlinear fiber is enough to produce supercontinuum generation. In previous FWM experiments [4], up to 6 kW peak power was available in a 600 ps long pulse from a Q-switched laser. In this case, a few metres of PCF provided efficient FWM. Since the maximum peak power available here is limited to 500 W, it is necessary to use a longer fiber.

In this section we analyze the spectra obtained using a 10 m long fiber sample. Figure 4 shows the evolution of the spectra emitted from the PCF as a function of input pump power. The growth of the idler and signal at 894 nm and 1315 nm respectively is in good agreement with the predictions of the phase matching diagram in Fig. 3. The FWHM bandwidths of the idler and signal are, respectively, 1 nm and 2 nm, and are independent of the pump power. The growth of the Raman shifted signal at 1120 nm is also observed in Fig. 4. The Raman signal is amplified exponentially from spontaneous emission, increasing as *exp(g _{R}P_{0}L/A_{eff})*, where

*P*is the peak power of the pulse,

_{0}*L*is the fiber length (the walk off distance of the Raman pulse is larger than the fiber length) and

*A*is the effective area of the guided mode. Figure 4 shows that the Raman signal increases by 5 dB for an increase of 70 W of peak power. The Raman gain,

_{eff}*g*, is 10

_{R}^{–13}m/W [15]. Taking this value into account, a value for

*A*of 3.8×10

_{eff}^{-11}m

^{2}is obtained, which is in good agreement with the calculated value. By using Eq. (3), a value of 0.0047 (mW)

^{-1}is obtained for

*γ*.

The FWM signal begins as spontaneous emission which is then amplified as *exp(γP _{0}L_{eff}*).

*L*represents the length of interaction between the pump pulse and the FWM signals and is effectively the walk off distance; dispersion implies that the FWM pulses and the pump pulse propagate at different speeds. Using the growth of the FWM signal shown in Fig. 4, an effective length (

_{eff}*L*) of 6 m is obtained. It is possible to calculate

_{eff}*L*theoretically from the dispersion properties of the fiber and, assuming a 27 ps pump pulse, a walk-off distance of 5 m is obtained, in good agreement with the experimental observations.

_{eff}The FWM signals dominate the Raman signal in the 10 m long fiber. A conversion efficiency, which is defined as the ratio of the summed average idler and signal powers divided by the output pump power, of 0.3% is obtained. The idler and signal grow exponentially when the distance is smaller than *L _{eff},* and linearly for longer fibers. In order to increase the conversion efficiency, it is desirable to use longer lengths of fibers. However, the walk-off distance for the Raman pulse is much larger than

*L*(a value of 30 m is calculated using the dispersion characteristics). Therefore, one can expect the Raman influence to become larger for longer fibers.

_{eff}## 4. Supercontinuum generation in 150 m long fiber

The full 150 m length of fiber is used in this set of experiments. Figure 5 shows the measured spectra for different pump power levels. Note that the global loss level at a wavelength of 1 µm is about 3 dB over 150 m of fiber.

The main feature of the spectrum up to 0.3 W of average output power is the growth of the Raman signal located at around 1120 nm. The critical peak power threshold can be calculated using the fiber parameters in the CW case [15, p302]. A value of 40 W for the peak power is found, corresponding to an output power of 0.3 W, in good agreement with the results of Fig. 5. The FWHM bandwidth of the Raman peak is about 10 nm when the average output power is below 0.3 W and increases dramatically for larger powers, leading to supercontinuum generation. Since the ZGVD wavelength is at 1070 nm, the pump wavelength lies in the normal dispersion regime while the Raman shifted wavelength is in the anomalous dispersion regime. At low pump power, the Raman signal intensity increases with an undistorted spectrum shape with a bandwidth of about 10 nm. The Raman peak bandwidth increases dramatically from an output power of 0.3 W to an output power of 0.48 W. It is a clear indication that a soliton has been formed. The pulse duration should be much shorter than the 30 ps pump pulse duration due to pulse compression. A similar situation appeared in [16], where a Nd:YAG delivering 90 ps pulses at 1.3 µm was used to pump a fiber in the normal dispersion regime while the Raman shifted signal was in the anomalous dispersion regime. The Raman pulse duration was measured to be 280 fs. If we assume here that the Raman pulse has a duration of, say, 500 fs, the corresponding peak power from the spectrum of Fig. 5 with 0.3 W output power is about 20 W. The corresponding soliton number, *N*, is about 3. This analysis is not rigorous since the pulse duration of the Raman pulse is unknown but it shows that the power in the Raman pulse is of the order of the power necessary to create a fundamental soliton. For larger pump powers, the intensity of the Raman signal does not increase much further and the energy is redistributed to longer wavelengths. Soliton dynamics lead to a very flat supercontinuum spectrum being formed, with a 1 dB variation from 1.15 µm to 1.35 µm for the flattest part. It is obvious from Fig. 5 (see curve corresponding to 1.14 W) that the strong OH absorption at 1.4 µm limits further expansion of the supercontinuum towards longer wavelengths.

The FWM signal can be seen for output powers larger than 1.5 W but the conversion efficiency is limited to 2%. Raman growth and supercontinuum generation largely dominate in this long fiber. The situation is the opposite to that of the 10 m long fiber. Again, the walk-off distance of the Raman pulse is much longer than that of the FWM signal (30 m versus 6 m), which explains the faster growth of the Raman signal. The conversion efficiency from the pump into the supercontinuum is about 90%.

## 5. Optimization of the conversion efficiency by varying the fiber length

In the previous sections the FWM signals grew faster than the Raman signal in a fiber whose length was comparable to the walk-off distance between the pump and the signal or idler. The contrary happens when the fiber is much larger than this walk-off distance.

Our aim is to optimize the efficiency of the FWM process. For this purpose, we propose to find the optimum length for which the efficiency is the largest. In addition, it so happens that the wavelength of the signal at 1.31 µm is a standard wavelength for diode lasers. Such a diode was used as a seed for the FWM amplifier to further improve the conversion efficiency, and a few tens of microwatts were coupled into the PCF. Figure 6 shows that the seed increases the FWM signal and therefore the efficiency typically by a factor of 10.

The results of pumping different lengths of fiber at maximum power whilst seeded are displayed in Fig. 7. For calibration purposes, the output light from the PCF was dispersed using a grating and the output powers at the pump wavelength, the signal and the idler wavelengths were measured using a power meter. The results were in good agreement with the corresponding integration of the spectrum from the OSA. Also note that the spectra from the Ando Model AO6317C OSA show an artifact peak at 840 nm (clearly visible on the top two spectra of Fig 7). This peak is visible even when a very weak pump beam is launched into the fiber; non-linear effects are negligible and the spectrum only comprises the pump wavelength and this spurious artifact peak.

The conversion efficiency of the PCF reaches up to 25% for a length of 150 m. At this stage, the pump is almost entirely depleted. However, the combined output power of the idler and signal saturates at a value of 0.4 W to 0.5 W because of the losses in the fiber. The Raman/supercontinuum signal increases very quickly up to 30 m. For longer distances there is a roughly linear increase. This is in agreement with a calculated walk-off distance for the Raman pulse of 30 m.

As shown in section 3, when the length of the fiber is longer than the coherence length (*L _{eff}* is about 5 m), the FWM signals increase as a function of the peak power

*P*like

_{0}*exp(γP*. Table 1 shows that the FWM efficiency increases roughly linearly with the fiber length, apart from the higher than expected conversion efficiency of the 30 m long fiber due to a better than usual coupling efficiency of the pump into the fiber.

_{0}L_{eff})Typically the best fit for the efficiency (see Fig. 8) is:

The first term inside the bracket including the exponential gives the FWM efficiency for a length equal to *L _{eff}* (about 4%), while the linear term provides the contribution for lengths

*L*>

*L*. This confirms the intuitive idea that the FWM process has to rebuild on spontaneous emission once the walk-off distance is exceeded and therefore the fiber can be subdivided into sections of length

_{eff}*L*, each providing a similar FWM output.

_{eff}The pump depletion is weak for the 10 m long fiber but is almost complete for the 150 m fiber and it is surprising that Eq. (4) gives a fair approximation in the depleted pump regime. It has to be acknowledged that Eq. (4) is a crude phenomenological model. In a much more realistic model, it is necessary to propagate the non-linear equations of motion. A detailed analysis of degenerate FWM including self-phase modulation and cross phase modulation terms is given in [17] and [18]. The authors analyse the case where the pump powers are large enough to get into the pump depletion regime over a length of fiber smaller than *L _{eff}*. It is worth noting that the theory mainly predicts a periodic variation of the powers for the pump, signal and idler as a function of distance with a periodicity close to

*L*. In our case, pump depletion increases asymptotically with distance. It is possible that non-reversible phenomena like loss and supercontinuum generation (effects not taken into account in [17] and [18]) prevent the periodical flows back and forth of the pump.

_{eff}## 6. Direction dependent FWM generation.

In this set of experiments we successively launched the pump light at each end of the same 50 m long fiber sample. The FWM process should be independent of whichever end of the fiber the pump is launched into, providing that the same amount of pump light is coupled at each fiber end. As can be seen in Fig. 9, this is not what is observed; which is an interesting finding as optical components for which the spectral properties are directional are uncommon. From one end of the fiber (which will be referred to as end 1), the idler is located at 900 nm and has a narrow linewidth of 1 nm. When the fiber is pumped from the other side (end 2), the idler is located at 925 nm and has a broader linewidth of 15 nm. In both cases, the signals are embedded into a continuum so are difficult to identify. The diode seed has no influence when the fiber is pumped through end 2 because the signal is at ~1260 nm and the diode emits at 1310 nm. But still, FWM is achieved with an efficiency of 40% with 1 W of average power in the idler at 925 nm. For comparison, the conversion efficiency is only 3–4% when the fiber is pumped through end 1 without seed. The signal and idler bandwidths increase by a factor of ten when the fiber is pumped through end 2 compared to end 1. It is therefore plausible that this is the reason for the increases conversion efficiency.

The exact reason for these much broader spectral lines compared to those produced while pumping through end 1, is not completely clear. One possibility is that there is a change of core dimensions close to end 2 of the fiber, resulting in a broadened FWM gain for spontaneous emission. Modeling shows that a very small variation in core diameter modifies the phase matching diagram significantly. Therefore, pumping each end of a tapered fiber would indeed generate different wavelengths through the FWM process. Using our phase matching conditions model, a core diameter variation of just 50 nm over the length of the fiber is sufficient to explain the wavelength shift observed. The effect of core diameter fluctuations on parametric gain has been analysed in [19]. It was found that fluctuations of core diameter would reduce the parametric gain. However the authors considered a periodic variation of core size. In the present case, the core dimension variation seems to be monotonic and produces a large shift in frequency for the FWM signals.

## 7. Conclusions

In this study, we have discussed the optimisation of the parametric generation in a PCF pumped by a mode locked Nd:YVO_{4} laser emitting 30 ps pulses at 1.064 µm with 500 W peak power (coupled into the fiber) and 7 W of available output power. The peak power of the laser is not large enough to get good FWM conversion efficiencies for a fiber length smaller than the walk-off distance between the pump and the FWM signals (about 5 m). By using fibers longer than this walk-off distance, a regime was entered in which the FWM signals increased only linearly with distance while the Raman increased exponentially for fiber length up to 30 m, and linearly for longer fiber lengths. Although pumping in the normal dispersion regime, supercontinuum generation was observed. The reason for this is that the Raman shifted peak lies in the anomalous dispersion regime. The Raman peak increases with a 10 nm bandwidth with distance to the point where its peak power is large enough to create a fundamental soliton, at which point a very flat supercontinuum extending from 1100 nm to 1400 nm was observed. A fiber seed emitting at 1310 nm and coinciding with the signal wavelength was used to increase the FWM conversion efficiency. Fiber lengths ranging from 50 m to 150 m produced a combined output power of 0.5 W in the idler and signal. Finally, we discovered that the generated wavelengths through the FWM process are direction dependent, probably because of variations in the fiber core dimensions along the fiber. Due to the larger linewidth of the FWM signals, a higher conversion efficiency of 40% was obtained with 1 W of average power in the signal at 920 nm.

## Acknowledgments

L. Michaille acknowledges Cato Fagermo from Crystal Fibre A/S for the loan of the 150 m long PCF tested in this paper and for allowing us to cut it into pieces. QinetiQ acknowledges support for this work from UK Ministry of Defence funding.

This paper includes material © British Crown copyright — Dstl 2008 — published with the permission of Her Majesty’s Stationery Office.

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