1. Introduction

There has been much interest in recent years in nonlinear interactions in optical fibres. The low nonlinearity of silica glass is offset by long interaction lengths and high power density in fibre to yield spectacular nonlinear effects. For most nonlinear processes the physical fibre length can be made longer than the effective interaction length, which is governed by phasematching pulse broadening, walk-off and attenuation. In particular, the fibre dispersion plays a key role in short pulse propagation and in phase matching conditions for nonlinear processes.

In the spectral region beyond 1300 nm, where the material dispersion of silica glass is itself anomalous, fibres can be designed and made to have a modal dispersion which is normal or anomalous, with a zero dispersion at any given wavelength (for example the dispersion shifted fibres used in telecommunications systems). It is not possible, however, to move the zero dispersion wavelength, λ0, of a silica step-index single-mode optical fibre to wavelengths shorter than 1270 nm, the zero dispersion wavelength of bulk silica [1]. In photonic crystal fibres (PCFs)[2–5], however, it is possible to shift the zero dispersion wavelength of single mode fibres to much shorter wavelengths [1,6,7]. This was exploited to dramatic effect in supercontinuum generation in small-core, high-index contrast PCF with zero dispersion wavelengths in the region 580–900 nm pumped with modelocked Ti:sapphire lasers at 750–850 nm [7–12]. Though these fibres are typically not strictly single-mode, higher order modes are difficult to excite and are also not coupled to the fundamental mode by normal bending, so the fibres may be used as if single mode. More recent results have included longer pulses, up to 60 ps, from modelocked krypton, Ti:sapphire or Nd-doped lasers [13,14,15].

In this paper we consider not just strictly single-mode fibres, but also so-called endlessly single-mode fibres which support only one guided mode over all wavelengths [3,4]. Fibres are designed and fabricated with zero dispersion wavelengths close on either side of the wavelength of a Nd:YAG laser at 1064 nm. We investigate in detail modulation instability, supercontinuum generation and optical parametric generation and amplification in these fibres when pumped with µJ energies in 600 ps pulses at 1064 nm. The use of Q-switched nanosecond pulses is a significant departure from previous work with modelocked femtosecond and picosecond lasers [7–15]. The laser technology required for Q-switching is much simpler than mode-locking, enabling savings in size and cost. There are also many Nd- and and Yb-doped lasers in the target wavelength range 1040–1070 nm, which can be directly diode pumped, and are thus compact and efficient.

2. Theory

Most previous supercontinuum generation experiments have focused on the ultra-short pulse regime, with femtosecond pulses from modelocked lasers [7,8,10,11]. In that case, self-phase modulation, soliton effects and pulse walk-off are important considerations, and the propagation is described by the generalised nonlinear Schrödinger equation [8,10,12]. Here we consider much longer pulses, where the propagation can be considered as quasi-CW. Neither the effects of dI/dt at the edges of the pulse, nor pulse walk-off between different wavelengths, are significant. In this case the major nonlinear process is phasematched four-wave mixing (FWM), to generate sidebands spaced at equal frequency intervals from the pump[8,13,14,16]. Gain for these processes is provided by the nonlinear refractive index of silica, n₂=2×10^-20 m²/W. Phase matching and conservation of energy give the equations[17]

and

where k_j are the wavevectors (propagation constants) of the modes, and ω_j the frequencies, of the pump, signal and idler waves; P is the pump power (in the quasi-CW case the peak pump power); and γ is the nonlinear coefficient of the fibre,

where A_eff is the effective area of the fibre and λ is the pump wavelength. These phasematching conditions will give the wavelengths for peak gain in a given fibre, and will depend on the chromatic dispersion of the fibre. We can measure or calculate the dispersion for different fibres and hence calculate the phasematching conditions (1). From numerical modelling of PCFs (using a full vector numerical model based on the supercell plane-wave method [18]) we obtain the propagation constants, k_i, directly, which may then be applied to Eq. (1). For measurements we only know the group velocity dispersion, the second derivative of the propagation constant. It is usual to expand the dispersion curve (as a function of optical frequency) as a Taylor series with dispersion coefficients β_n, from which the phasematching (1) can be calculated [8,17]. For the PCFs considered here we included terms up to β₆ in order to provide a reasonable fit and extrapolation for the measured group velocity dispersion curves (Fig. 1(a)). The Taylor coefficient β ₂ (ps²/km) is related to the engineering unit for group velocity dispersion, D (ps/nm km), by

Phasematched FWM wavelengths calculated from the measured dispersion of one PCF are shown in Fig. 1(b) as a function of the pump wavelength offset from the zero dispersion wavelength, λ ₀. There are three important regions: a) λ _pump« λ ₀, b) λ _pump ≲ λ ₀, c) λ _pump › λ ₀.

 

Fig. 1. (a) Measured dispersion curves for fibres G, O and P, together with the dispersion calculated for a regular PCF with round holes and pitch, Λ, 3 µm and d/Λ=0.3. A dashed vertical line indicates the pump wavelength, 1.064 µm; (b) Nonlinear phasematching diagram for the process 2ω _pump → ω _signal + ω _idler, calculated from the measured dispersion curve of fibre G for input powers P=14 W (blue curve); P=140 W (green curve); P=1400 W (red curve). Circles: measured parametric wavelengths corresponding to pump wavelength offset from λ ₀ for fibres C, F, G, H, I, L (Table 1).

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Taking these in reverse order; case c) (the right hand half of Fig. 1(b)) shows a strongly power-dependent phasematching of FWM peaks close to the pump wavelength. A non-zero value of γP is required for solutions of (1) in this region. This is the well known phenomenon of modulation instability (MI) which occurs in the anomalous dispersion regime of all fibres. The gain peaks are relatively broad, and the central frequencies depend mostly on the group velocity dispersion, β ₂, and only weakly on the higher order dispersion.

In case b) (the left hand half of Fig. 1(b)) there is a largely power-independent phasematching of widely spaced FWM peaks. Solutions of (1) in this region are present even for zero power, but only for non-zero higher order dispersion (even terms only, β ₄, β ₆ etc, in the Taylor expansion [8]). The gain peaks are relatively narrow, and the central frequencies depend strongly on the higher order dispersion.

In case a) (beyond the left side of Fig. 1(b)) there is no phasematching for FWM. The boundary between a) and b) has an experimental and a theoretical position. It can be seen from Fig. 1(b) that the idler wavelength is shifted further beyond 2 µm as the offset of the pump from λ ₀ is increased. Idler signals generated beyond 2.2 µm cannot be detected because of the absorption of silica increases rapidly in this wavelength range. Even neglecting absorption, an idealised fibre shows FWM phasematching branches which curve back on themselves, giving a limit to the maximum wavelength offset at which FWM can occur.

Widely spaced FWM peaks (case b) have been discussed frequently [8,13,14,16], but were only recently observed, using 60 ps pulses at 647 nm from a modelocked Kr⁺ laser in a PCF with zero dispersion wavelength at 652 nm, by the current authors and others [19]. In this work we investigate the FWM/MI phenomena in greater detail, with pulses an order of magnitude longer, 600 ps, and at wavelength, 1064 nm, of great engineering importance, given the abundance of different Nd- and Yb-doped lasers available.

As well as FWM/MI gain, all silica fibres will display Raman gain, at the characteristic shift of 13 THz. As this is not a phasematched process, it will occur in all fibres and is largely unaffected by differences in the fibre dispersion. Where phasematching is available, FWM/MI gain is generally higher than Raman gain in silica, so significant Raman effects are only expected to be observed when FWM/MI gain is not present (i.e., for case a).

 

Fig. 2. SEM of fibre O. Λ=2.97, d/Λ=0.39, λ₀=1065 nm

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3. Experimental conditions

Many PCFs were fabricated with zero dispersion wavelengths to either side of 1064 nm. The fibres have a 125 µm outside diameter and 250 µm acrylate buffer for compatibility with standard fibre cleavers, strippers, mechanical holders and adaptors. All of the fibres have nominally the same hole-to-hole pitch, Λ=3 µm, but with different hole diameters, d, from d/Λ=0.3 to d/Λ=0.5, corresponding to a core diameter of approximately 5 µm. For larger holes the zero dispersion wavelength lies to shorter wavelengths. Measured zero dispersion wavelengths, λ ₀, span from 1040 nm to 1105 nm. No specific attempt was made during fabrication to reduce fibre losses, and as a consequence these were relatively high, being 4.5 dB/km at 1550 nm, and 12 dB/km at 1064 nm, with 110 dB/km at the peak of the OH absorption at 1380 nm. A scanning electron micrograph of a representative fibre is shown in Fig. 2. For comparison, a conventional step-index fibre, Nufern 1000-HP, which has a single-mode cut-off wavelength of 920 nm and mode-field diameter 6.2 µm at 1060 nm, was also investigated.

Nonlinear interactions in the fibres were observed by pumping with 600 ps pulses from a passively Q-switched Nd:YAG laser (JDS Uniphase model number NP-10620-100). The average power delivered to the fibre was 30 mW, with a pulse repetition rate of 7.25 kHz, corresponding to a pulse energy of 4.1 µJ and a peak power of 6.9 kW. Coupling efficiency into the various single-mode fibres was 35–50%. This pump laser is low-cost and extremely compact with a laser head 100×22×32 mm which adds a practical usefulness to the scientific interest in wavelength conversion and continuum generation. Power input to the fibre under test was controlled using a mica waveplate and crystal polarizer. The polarization of the input to the fibre was fixed to be vertical at all times. Input and output powers were measured with a thermal power meter because of its flat spectral response over the wide range of output wavelengths generated. Output spectra were measured with an optical spectrum analyser (Ando AO-6315B). The spectral resolution was set to 5 nm except where stated otherwise. Powers at discrete parametric wavelengths were measured by dispersing the output with an SF11 equilateral prism and measuring the individual beams with a thermal power meter. For measurement of parametric gain, the output from a fibre coupled CW diode laser was introduced into the input beam by reflection from an uncoated glass plate at 45°. The polarization of the diode was adjusted for maximum reflection from the plate, which corresponds to predominately vertical polarization, parallel to the pump light polarization. The seed power coupled into the fibre was measured at the fibre output using a low power photodiode detector calibrated at the seed wavelength.

Table 1. Parametric generation wavelengths for the fibres studied.

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λ ₀, measured zero dispersion wavelength (nm); λ _signal, measured OPG signal wavelength (nm);

λ _idler, measured OPG idler wavelength (nm) – shaded values calculated from λ _signal (nm).

4. Results

Table 1 shows the optical data for several PCFs considered in this paper. The dispersion was measured using a low-coherence interferometric technique[20]. The optical parametric generation (OPG) wavelengths refer to the measured output wavelengths when a short, 1 to 3 m, length of fibre is pumped with pulses at 1064 nm. All the fibres listed in Table 1, except for fibre P, are endlessly single mode; there is only one guided mode whatever the wavelength. Fibre P which has hole diameter d/Λ>0.4 is not endlessly single mode, however the single mode cut-off wavelength is <650 nm, so it is single mode at the wavelengths of interest. Measured dispersion curves for a selection of the fibres are shown in Fig. 1(a), together with the curve calculated for an idealised fibre with Λ=3 µm d/Λ=0.3. The different regimes of nonlinear interaction described in section 2 are all accessible with the range of fibres available; a) λ _pump « λ ₀, as represented by the Nufern 1000-HP conventional step-index fibre, b) λ _pump ≲ λ ₀, as represented by PCF L, c) λ _pump › λ ₀, as represented by PCF P. For each case the evolution of the output spectrum with input power and fibre length is discussed in the sections below:

4.1 Case a) λ_pump « λ₀

The step-index fibre 1000-HP has a measured zero dispersion wavelength λ ₀=1440 nm. The pump wavelength offset is very large, -376 nm, which lies in the region where there is no nonlinear phasematching. The dispersion at the pump wavelength, 1064 nm, is -37 ps/nm km. The evolution of the measured output spectrum for 100 m of this fibre with input power is shown in Fig. 3. There is significant Raman generation, with several orders of Raman Stokes lines visible. The spectrum is one-sided, with no generation of wavelengths shorter than the pump. This is a clear indication of the absence of parametric processes, as is expected.

 

Fig. 3. Measured output continuum spectra from 100 m of Nufern 1000-HP single mode fibre. False colour scale in dBm/5nm bandwidth.

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4.2 Case b) λ_pump ≲ λ₀

PCF L has a measured zero dispersion wavelength λ ₀=1069 nm. The pump wavelength offset is small, -5 nm, which lies in the region where there is phasematching of widely spaced wavelengths, with little power dependence (FWM, the left half of Fig. 1(b)). The dispersion at the pump wavelength is also small, just -1 ps/nm km. The evolution of the measured output spectrum with input power is shown in Fig. 4(a) for 6 m of this fibre. At low power, two distinct parametric wavelengths are generated at 895 and 1315 nm, equally spaced in energy about the pump wavelength. This is as expected from phasematching calculations. As the pump power is increased further, there is spectral broadening about the pump, signal and idler wavelengths. For other PCFs, A-N, with the pump offset from λ ₀ by up to -40 nm, similar parametric generation is seen, with signal wavelengths ranging from 686 nm to 975 nm, and idler wavelengths ranging from 1168 nm to beyond 1900 nm (Table 1, Fig. 5(a)).

The broadening of the generated parametric peaks seen at high power in Fig. 4(a) is considerably reduced for fibres generating more widely-spaced FWM wavelengths. For example, Fig. 5(b) shows the output for fibre B. Here there is very little broadening of the pump and signal wavelengths as the pump power is increased. To understand this we must consider not only the exact phasematching given by solutions of Eq. (1), but also the gain at signal wavelengths close to solutions of (1). When the signal and idler are close to the pump wavelength (as for PCF L), there is a slow variation of phase mismatch around the exact phasematching condition and the gain bandwidth is correspondingly wide (see Fig. 3 of Ref. [8] for a full gain calculation in similar conditions). When the signal and idler are far from the pump wavelength, however, (as for PCF B) the phase mismatch changes very rapidly close to the exact phasematching condition. This yields a correspondingly narrow parametric gain peak. The relative widths of the phasematching peaks can also be see in the power dependence of solutions of Eq. (1) shown in Fig. 1(b). The power term, 2γP, in Eq. (1) has little effect on solutions at the left-hand side of the plot, where the signal and idler are far from the pump wavelength, but has a large effect on the right-hand side of the plot where the signal and idler are close to the pump wavelength.

 

Fig. 4. (a) Output spectra for 6m length of PCF L showing strong optical parametric generation in the normal dispersion regime; (b) Signal output for a 2.5 m length of PCF L with 2mW pump and 9.5 (blue), 4.2 (red), 1.4 (purple), 0.07 (green) µW seed. Pump only, no seed, black. 1µW cw seed is 4000 photons in 600 ps. Spectrometer resolution 0.1 nm.

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Fig. 5. (a) Output spectra for 3m lengths PCFs A, C, F, G, H, I showing strong optical parametric generation in the normal dispersion regime, input power 10–20 mW. Spectrometer resolution 0.2 nm. Idler wavelengths longer than 1750 nm are not measured with this spectrometer. (b) Power dependence of spectra for fibre B generating λ _signal=716 nm.

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The spectrum of the 716 nm signal is shown in Fig. 6 for medium and high input power, together with the evolution of the bandwidth of the 716 nm and 1064 peaks with pump power. The bandwidths are unchanged for pump powers up to 25 mW, when both increase to 1.8 nm FWHM at 30 mW pump power. The parametric conversion efficiency in this fibre was determined by measuring the power of the signal and pump beams dispersed by a prism. For 30 mW input power, the total output was 11 mW, of which 8.3 mW was pump at 1064 nm and 2.5 mW was signal at 716 nm, a conversion of 22%. No radiation was measured at the expected idler wavelength of 2.07 µm. We believe that confinement loss at long wavelengths is the reason for the absence of this wavelength in the output. Using fibre C with a smaller pump wavelength offset, the FWM wavelengths are slightly closer at 732 nm (measured) and 1945 nm (inferred from the signal wavelength). In this case, output radiation at the idler wavelength was observed. For a 3 m length at a pump power of 30 mW, the total output power was 13 mW, of which 8.0 mW was pump at 1064 nm; 4.5 mW was signal at 732 nm, a conversion of 35%; and 0.43 mW was idler at 1945 nm, a conversion of 3%.

The wavelengths of parametric generation measured in fibres C, F, G, H, I, L are plotted in Fig. 1(b) against the pump wavelength offset from the measured λ ₀ for each fibre. Good agreement is seen between these points and the lines calculated by Eqs (1) and 2 from the measured dispersion of fibre G.

Parametric gain at 1315 nm was measured for a 2.5 m length of fibre L using a CW diode laser probe beam. At a coupled pump power of 4 mW (peak power 920 W), where the spontaneous parametric generation is still low, a gain of >55 dB was measured for a seed power of 15 µW at 1315 nm. The probe laser operates on many longitudinal modes (Fig. 7), is not actively stabilised and the duty cycle of the pump laser is very low, all of which make it difficult to measure the gain of the probe accurately from CW measurements. Figure 7 shows that the amplified probe diode modes rise at least 20 dB above the spontaneous background, and have a gain of a factor of 3–4 over the un-amplified probe output. Using the pump pulse duty factor, this observed increase corresponds to a pulsed gain with a conservative lower limit of 55 dB.

The threshold for observation of light at the signal wavelength (895 nm) was lowered from 2 mW (460 W peak) pump power for unseeded spontaneous generation, to 0.95 mW (218 W peak) for a seed power of 11 µW at 1315 nm. At 2 mW (460 W peak) pump power, seeded parametric generation was observed for the lowest achievable seed power of 0.07 µW, which corresponds to fewer than 300 photons during the 600 ps gain period, Fig. 4(b). The modes seen in the seeded signal correspond to the longitudinal modes of the seed diode laser at 1310 nm. The pump and seed powers required are sufficiently low that one might reasonably expect to generate CW parametric oscillation using a longer fibre with feedback.

 

Fig. 6. Detail of spectra from Fig. 5(b), fibre B. Spectrometer resolution 0.2 nm. (a) Line width (full width half maximum) of the output at the pump wavelength (1064 nm, red dashed line) and the OPG signal wavelength (716 nm, green solid line). (b) Normalised output spectra at the OPG signal wavelength for low and high input powers.

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Fig. 7. Example output spectra for measurement of gain in 2.5 m of PCF L using a CW diode probe. Pump power 4 mW (920 W peak) at 1064 nm, seed power 15 µW at 1315 nm. Green trace, pump only; red trace, probe only; blue trace, pump and probe. Spectrometer resolution 0.1 nm.

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4.3 Case c) λ_pump › λ₀

PCF P has a measured zero dispersion wavelength λ ₀=1039 nm. The pump wavelength offset is +25 nm, which lies in the region where there is power-dependent phasematching of closely spaced wavelengths (MI, the right half of Fig. 1(b)). The dispersion at the pump wavelength is +5 ps/nm km. The evolution of the measured output spectra for 1 m, 3 m, 20 m and 100 m of this fibre with input power is shown in Fig. 8. For short 1 m and 3m lengths, the symmetrical MI peaks are clearly visible close on either side of the pump wavelength. At low power (5–7 mW for a 1 m length) there is a shift of the generated MI wavelengths with input power as expected from equation (1), but once there is significant power in the MI peaks the wavelengths become fixed through saturation, and new higher-order MI sidebands appear. The new pump wavelengths in these MI sidebands yield many new phasematched solutions of equation (1) which combine to yield a broad continuum spectrum. For long 20 m and 100 m lengths of fibre, defined MI peaks are only visible at the very lowest powers, < 2 mW. The wavelengths generated are much closer to the pump (scarcely separated from the pump for 100 m), as expected from the lower pump power at which they are observed, and again the positions of the peaks stabilise at high power. At high power the output bandwidth grows into a broad and extremely flat continuum, spanning from approximately 500 nm to beyond the limit of the OSA at 1750 nm. Other detectors were used to show that there is certainly power in the spectrum beyond 1900 nm. Representative high power spectra for 20 m lengths of two fibres are shown in Fig. 9 on both linear and logarithmic scales. The lack of spectral features in the flat continuum is in marked contrast to continua generated in PCF with femtosecond pulses[7,10,11]. Short and medium-term temporal stability is also good, as we have applied this continuum as a source for interferometric measurements without the need to monitor the input power.

As the spectrum is already extremely broad after 20 m of fibre, little bandwidth is gained from further propagation to 100 m. In fact, the main effect of further propagation is power loss. The propagation is not, however, passive linear propagation of the broad spectrum generated in the first 20 m of fibre. This can be seen by looking at the dip in the output spectrum caused by the OH absorption of the fibre at 1380 nm which amounts to 8 dB for passive fibre propagation from 20 to 100 m. The actual dip measured in the spectrum after 100 m is only 4 dB, suggesting that there is sufficient power in the continuum on either side of the absorption to be able to continue to re-distribute energy into this region as energy is lost to absorption.

 

Fig. 8. Measured output continuum spectra from (a) 1 m, (b) 3m, (c) 20 m and (d) 100 m of fibre P. False colour scale in dBm/5nm bandwidth.

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Fig. 9. Output spectra for 20 m lengths of fibres at 30 mW input power. (a) logarithmic scale, fibre O (red trace) and P (blue trace), (b) linear scale, fibre P (arbitrary units, normalised to residual pump peak at 1064 nm).

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5. Conclusion

We have demonstrated a new dispersion regime for single mode fibres, where the zero-GVD wavelength is close to 1064 nm. This is applied to nonlinear interactions of sub-nanosecond Q-switched laser pulses, either to produce a compact source of broad, flat, spectrally and spatially bright single mode continuum radiation, or for compact efficient wavelength conversion to produce pulses at a selected wavelength in the near IR.

A broad, flat and compact continuum source has obvious application to spectral testing of fibre components (for which it already proving its power and versatility in our laboratory) and spectral analysis of chemical and biological samples. Pulsed narrow band sources at other selected wavelengths increase the range of wavelengths easily available for nonlinear identification and detection in schemes such as two-photon fluorescence, as well as providing pump sources for nonlinear interactions in fibres at other interesting wavelengths. For example, pulses generated at 750 nm could be launched into nonlinear dispersion shifted PCF designed for continuum generation with Ti:sapphire lasers[7,11], and yield a continuum spanning further into the visible than is possible when starting at 1064 nm in the IR.

The observed nonlinearities fit to well understood physical processes of FWM and MI, and the control of dispersion readily available with PCF technology[5] has enabled application to a wavelength of great importance in laser engineering. Further consideration of fibre dispersion may help to improve further on the results presented here.