## Abstract

We report in detail, both experimentally and using numerical simulation, the efficiency of generation of supercontinua in optical fiber driven by modulation instability of a continuous-wave (CW) pump source. It is shown that the degree of pump coherence has a dramatic effect on the resulting spectral expansion and it is discussed how this can be explained by having the proper conditions for efficient modulation instability to break the CW pump light into a train of fundamental solitons that subsequently undergo self-Raman shift to longer wavelengths. It is proposed that an optimal pump bandwidth exists corresponding to the optimal degree of pump incoherence, defined as a function of the modulation instability period.

© 2012 Optical Society of America

## 1. INTRODUCTION

Supercontinuum generation in an optical fiber, pumped with a relatively low-power (typically tens of watts) continuous-wave (CW) source, has proven to produce the flattest spectra and highest spectral powers of all pump configurations [1–13]. Modulation instability (MI) is the key mechanism that initiates the formation of high peak power temporal optical solitons from a low-power CW field, a process inherent to any anomalously dispersive nonlinear medium [14]. It is the subsequent redshifting of the solitons generated from MI, through self-Raman interaction, influenced by their local dispersion and collision events between temporally and spectrally coincident solitons, that gives rise to the broad spanning spectra of long-pulse and CW-pumped supercontinua [15,16]. The availability of efficient, high-power CW-fiber lasers and specialty optical fibers, with highly engineered dispersion profiles, has facilitated all-fiber integrated supercontinuum light sources with spectral powers approaching $100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mW}\text{\hspace{0.17em}}{\mathrm{nm}}^{-1}$, filling the transmission window of silica [8,11].

Without special attention paid to the modal content, it is typical for fiber lasers to have thousands of oscillating longitudinal modes [17–19]. Without the inclusion of a mode-locking element, providing a fixed phase relationship between the longitudinal modes, strong stochastic intensity fluctuations will be present in the laser’s output. It has been shown previously that pump noise arising from such intensity fluctuations in a CW laser can strongly influence the shape of the resulting supercontinuum [20]. The average duration of these fluctuations, or the coherence time ${\tau}_{c}$, is inversely proportional to the pump bandwidth: the faster the decorrelation of the wave (i.e., smaller ${\tau}_{c}$), the larger the range of frequencies the wave contains, and the broader the spectral linewidth [21]. Here, we study the effect of the pump coherence on the MI gain and show how this influences the resulting spectral expansion in the evolution of a supercontinuum generated in a length of highly nonlinear fiber (HNLF).

The structure of the paper is as follows. In the following section we provide a brief overview of the theory of CW supercontinuum generation (SCG), then outline our hypothesis: how pump incoherence affects the MI and, hence, the CW continuum formation process. In Section 3 we present the experimental scheme consisting of a CW pump system with a broadly tunable temporal coherence, followed by a length of HNLF. The construction of a suitable numerical model to analyze the experimental configuration is developed in Section 4, consisting of two components: a model to describe the CW laser, and a model to propagate the laser field in the HNLF. The numerical and experimental results are presented and discussed in Section 5 and Section 6, and, finally, conclusions drawn in Section 7.

## 2. THEORY

#### A. Basic Theory of CW Continuum Generation

The general mechanism and detailed dynamics of the CW continuum formation process have been studied in great detail [13]. Here, we review the essential points useful for the rest of this paper.

CW supercontinuum formation can be broken into three clear stages [22].

- 1. The breakup of the input pump source into solitons due to MI, by pumping in the anomalous dispersion region of a nonlinear fiber.
- 2. The redshifting of the solitons though Raman self-frequency shift and Raman mediated soliton collisions.
- 3. If MI-induced solitons form sufficiently close to the zero dispersion wavelength (ZDW), they can excite dispersive waves in the normal dispersion region. These can then be further blueshifted due to the trapping of dispersive waves by redshifting solitons.

Stage 1 is characterized by the MI period ${T}_{\mathrm{MI}}$, given by [23]

The MI period is an important parameter for a number of reasons: it defines the number of solitons that are generated per unit of time; with one soliton generated per period, under conditions of energy conservation, ${T}_{\mathrm{MI}}$ can be used to calculate the energy of each soliton emitted from MI [13]:

From Eq. (2) it is clear that, as the MI period reduces, or the MI bandwidth increases, the soliton duration reduces and the corresponding soliton peak power, ${P}_{0}$, increases. The duration of solitons emitted from MI can be estimated by considering the relationship between ${\tau}_{0}$ and ${P}_{0}$ given by the soliton equation [23] such that Using Eq. (1) in Eq. (4) and rearranging for ${\tau}_{0}$ gives Thus the estimated full width at half-maximum duration (FWHM) of a soliton emitted from MI is approximately ${T}_{\mathrm{MI}}/5$ [13,24].The extent to which stage 2 occurs depends on the duration of the solitons emitted from the MI process. Shorter solitons have broader bandwidths, which must be sufficiently broad for a significant fraction to overlap with the Raman gain spectrum, to undergo self-scattering (otherwise known as the soliton self-frequency shift). Although MI initiates the spectral broadening in CW SCG (where the peak powers are typically $<100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{W}$), it is subsequent soliton collisions, and primarily the Raman scattering of solitons formed from MI, that provide the majority of the spectral expansion [15,25]. The rate at which the solitons Raman shift is proportional to the inverse of the fourth order of their duration, such that

In addition, because ${T}_{\mathrm{MI}}$ is not a fixed quantity when evolving from noise—this is clearly seen in frequency space with the formation of symmetric sidebands either side of the pump frequency, with maxima at ${\omega}_{0}\pm \mathrm{\Delta}{\omega}_{\text{max}}$, but with a linewidth indicating that the MI modulation period is not precisely fixed—solitons are emitted with a distribution of durations, consequently Raman shifting to different positions in frequency space [26,27]. It is this variation in the soliton duration that causes the smooth average spectra expected of CW supercontinua. Despite having a high degree of spectral flatness, this also results in low temporal coherence.

#### B. Designing Continuous-Wave Supercontinuum Systems

The above analysis can assist in the design of experimental systems for CW supercontinuum. There are three competing demands to be met [13].

- 1. The ratio $|{\beta}_{2}|/\gamma $ should be reduced as much as possible, as this both increases the efficiency of the MI process, and also leads to the generation of the shortest possible solitons, leading to maximum Raman self-interaction and maximum redshift.
- 2. However, the dispersion slope (and to a lesser extent, the reduction of $\gamma $) should be minimal or negative with increasing frequency, so that solitons undergoing Raman self-frequency shift obtain the maximum redshift before they adiabatically broaden to a point that prevents further redshift.
- 3. For blueshifted CW supercontinua, the MI-induced solitons should propagate close to a zero dispersion point, to generate dispersive waves.

Note that point 1 is often in conflict with point 2; i.e., reducing $|{\beta}_{2}|$ at the pump wavelength often suggests one should move close to the first ZDW, as does point 3. However, this can prevent any significant redshifted continuum from forming due to the consequent large positive dispersion slope. This is unavoidable if one wishes to obtain blueshifted or visible CW supercontinua; but it has been found that a higher absolute value of $|{\beta}_{2}|$ in a fiber with a very flat dispersion profile is optimal for large redshifted CW continuum formation [7,10,11,13].

#### C. Role of Pump Coherence in MI

It is well known that, for both instantaneous and noninstantaneous nonlinear media, wave incoherence (or partial coherence) affects the efficiency of modulation instability gain [28–30]. One model for optical fiber suggests the following modification to the purely CW MI gain term in the presence of incoherence:

Two conclusions can be drawn directly from Eq. (7): first, that the MI gain strictly decreases with increasing pump bandwidth (or increasing incoherence of the pump wave); in fact, MI gain is eliminated completely when the pump bandwidth is equivalent to the MI bandwidth. The second conclusion is that the maximum MI gain occurs for the most coherent pump source. Oddly, in the limit of infinitesimal pump bandwidth, Eq. (7) suggests that the MI gain is still higher than that derived for the coherent MI case.

In contrast, it has been experimentally observed that broader pump bandwidths, at least initially, produce more efficient supercontinua [20], and can be expected to enhance the MI gain [13,30], through the enhancement of instantaneous peak powers in the partially coherent pump source. We suggest that this discrepancy with Eq. (7) is due to the fact that the increased peak power fluctuations are not fully accounted for in that equation.

The temporal coherence (${\tau}_{c}$) of a light source, with a complex electric field $E(t)$ can be determined by considering the field autocorrelation (AC) given by

*second-order coherence function*[31], which is the Fourier transform of the spectral power $S(\omega )$ [21]. This relation clearly links the pump spectral bandwidth inversely with the coherence time. However, it says nothing about the temporal intensity fluctuations; all of the spectral width could be (momentarily) due to phase fluctuations.

The intensity AC of a complex, or highly structured waveform, such as a CW laser, given by [31]

We therefore conclude that the field AC is redundant if we measure the spectrum, which provides identical information, whereas the intensity AC characterizes the intensity modulations present on the background CW signal. In order to quantify both the coherence time and intensity fluctuations of our experimental system as a function of the bandwidth, we measured the spectrum and background-free intensity AC.

Figure 1 shows the simulated temporal and spectral field intensities for three pump bandwidths using the model described in more detail in Section 4. It is clear from Fig. 1 that, as the spectral bandwidth of the CW pump source increases, the rate of fluctuations in the temporal domain increases: the decorrelation time of the wave increases or the temporal coherence of the field, ${\tau}_{c}$ decreases. It is also evident that, as ${\tau}_{c}$ decreases, the associated peak power increases. However, the effect of increased instantaneous power saturates at a certain pump bandwidth. This is confirmed by the clear logarithmic dependence of the peak power enhancement factor, ${\mathrm{\Psi}}_{\text{enhancement}}$ (defined as the ratio of the peak and average power), on the pump linewidth shown in Fig. 2. Because of the stochastic nature of the initial noise conditions, as with all simulations performed for this paper, the peak power enhancement factor was averaged over an ensemble of simulations, each seeded from a different random initial noise condition, for each pump bandwidth. The five-point averaged data are shown in Fig. 2 with blue dots, and the logarithmic fit to this averaged data with a solid blue curve. The corresponding coherence time, ${\tau}_{c}$, is also plotted, showing an inverse log–log dependence on the pump bandwidth: as the linewidth of the pump laser increases from 0.1 to 10 nm, the coherence time decreases by 2 orders of magnitude from 100 to 1 ps.

We suggest that the optimal pump bandwidth for efficient production of short solitons through MI, and, hence, efficient CW SCG, is not the nearly coherent pump suggested by Eq. (7), but much broader. For very narrow pump bandwidths, the coherence time of the pump source is much longer that the MI period; therefore, the peak MI gain is determined by the peak power fluctuations of the pump source. These peak power fluctuations increase as the bandwidth is moderately increased and the coherence time decreased, hence, the MI and continuum efficiency should also increase. When the bandwidth increases so much that the pump coherence time is reduced below the MI period, i.e., the pump bandwidth becomes comparable to the MI bandwidth, the gain is reduced by a corresponding amount, and eventually is completely inhibited.

In the following sections we provide experimental and numerical support for this hypothesis.

## 3. EXPERIMENTAL SETUP

#### A. Broadly Tunable Amplified Spontaneous Emission Source

An overview of the laser pump system used in our experiments is shown in Fig. 3(a). It comprises a chain of two low-power Er-doped fiber amplifiers generating amplified spontaneous emission (ASE) within their gain bandwidth (1545–1575 nm), intersected by a broad, but fixed bandwidth (12 nm FWHM passband), bandpass filter, centered at 1565 nm. The filter acts to increase the signal-to-noise ratio, by suppressing ASE outside the desired bandwidth. This assembly forms the seed for a 10 W Er-doped power amplifier. A tunable bandwidth filter is employed to control the linewidth of the seed source passing into the power amplifier, allowing continuous control of the pump source bandwidth from $\sim 0.1\u20137.0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, corresponding to a coherence time range of $\sim 20\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}\ge {\tau}_{c}\ge 50\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fs}$. The lower limit on ${\tau}_{c}$ is restricted by the imprint of the finite, parabolic gain shaping of successive stages of amplification in the Er-doped fiber amplifiers and not by the maximum allowable bandwidth of the tunable filter.

The spectral and temporal performance of the pump system is shown in Fig. 4, where the FWHM spectral bandwidths were 0.36, 1.77, and 5.28 nm, for Figs. 4(a) and 4(d), 4(b) and 4(e), and 4(c) and 4(f), respectively. Greater than 30 dB suppression between the signal and pedestal was achieved. Figure 4(c) clearly shows the affect of multiple stages of amplification and the shape of the gain profile of the amplifiers imposed on the output spectrum—ultimately limiting the minimum coherence time available from this system.

The corresponding AC function for the pump bandwidth shown in Figs. 4(a)–(c) is given in Figs. 4(d)–(f), respectively. A two-to-one contrast between the peak and the pedestal of the function shows that the field consists of 100% modulations, where the average duration of the modulation is given by the width of the coherence spike. As such, it is clear to see that, for the narrowest pump bandwidth, there are fluctuations on the time scale of tens of picoseconds, reducing to tens of femtoseconds as the bandwidth increases.

#### B. HNLF Details

The output of the tunable ASE source was directly fusion spliced to a high-power, inline, fiber pigtailed optical isolator to prevent spurious backreflections from damaging upstream components, due to the high gain of the final stage amplifier. The output of the isolator was fusion spliced directly to a 50 m length of HNLF [see Fig. 3(b)]. The splice was optimized on an arc-discharge fusion splicer using a mode-matching algorithm, with repeatable splice losses of $\sim 0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$. The details of the HNLF are summarized in Fig. 5(a): the calculated ZDW is $\sim 1.47\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, a second ZDW exists at 2.14 μm; the calculated dispersion at the pump wavelength is $2.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}\text{\hspace{0.17em}}{\mathrm{nm}}^{-1}\text{\hspace{0.17em}}{\mathrm{km}}^{-1}$, and the calculated nonlinear coefficient at the pump wavelength is $9.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{W}}^{-1}\text{\hspace{0.17em}}{\mathrm{km}}^{-1}$.

The MI period for the fiber used in our experiments is plotted in Fig. 5(b), as a function of the average pump power at a fixed pump wavelength of 1.565 μm. At the average power of 6.3 W (typical for all results presented here), ${T}_{\mathrm{MI}}\approx 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$.

## 4. CONSTRUCTING A NUMERICAL MODEL

#### A. Modeling a CW Pump Source

In order to investigate the role of pump source coherence in CW-pumped SCG an appropriate model of the pump system is a prerequisite for use as the initial condition. A suitable numerical model of a CW-fiber-based source containing empirically valid fluctuations in the amplitude and phase of the field is a complex problem, and a number of models have been proposed [18,19,26,32–34]. In addition to difficulties regarding physical initial conditions, CW supercontinuum simulations are necessarily going to be an approximation due to the finite periodic boundary conditions imposed to make the problem tractable.

Considerable simplification is possible when using a correctly isolated cascaded ASE source, as the entire experimental system is strictly forward propagating, eliminating the need to model cavity mode effects, and allowing us to use standard forward propagating generalized nonlinear Schrödinger equation (GNLSE) models to simulate the nonlinear evolution through the amplifier systems. It should be noted that, given that ASE- and CW-laser-based supercontinua are essentially identical if the pump bandwidths are matched [4], this model should be transferable to laser pump cases.

We modeled our experimental pump system [which comprises a chain of erbium amplifiers and tunable filters; see Fig. 3(a)] by iterating an initial white noise field, equivalent to one photon per mode [35], through an amplifier stage, followed by a spectral filtering element, until the average power and spectral width match our pump conditions. Examples of the temporal and spectral intensity profiles output from this model are shown in Fig. 1.

In order to maintain a constant average power, it is necessary to adjust the pump power as the filter bandwidth is increased, as filtering out less power results in a lower insertion loss. In the laboratory, a constant launched pump power was maintained by simply adjusting the pump power of the final amplifier stage as the filter bandwidth was varied, to obtain the required average output power. In our numerical model, we took a similar approach using a simple proportional, integral, differential (PID) feedback control loop, implemented in software, to iteratively find the correct value of amplifier saturation energy (analogous to our pump power control in the laboratory) for a given pump bandwidth, to within 1% of a target time-averaged power.

Using an ASE-based source means that the temporal coherence of the pump system can be broadly tuned, within the gain bandwidth of the amplifiers used, with a high degree of control by employing a tunable bandpass filter (TBPF) after a final preamplifier stage and before a power amplifier, while ensuring all other parameters of the pump system remain unchanged (i.e., average launched power, central wavelength, etc.). This is important for isolating parameters that affect the evolution, and ultimately the optimization, of the SCG. Here, we use a TPBF with a passband tunable from 0.1 to 15 nm, centered around 1565 nm, and compatible with high-power fiber-based Er amplifiers. However, given the finite gain bandwidth of Er-doped fiber amplifiers (approximately 30 nm), it is only possible to utilize in the ideal case a FWHM pump bandwidth of $\sim 10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ because of the cascaded single-pass gain shaping effect of the amplifier chain. This is illustrated in Fig. 6, where the calculated FWHM pump bandwidth is plotted as a function of the TBPF bandwidth. The data was computed using our numerical model of the experimental pump system, with a gain bandwidth of 30 nm specified for all amplifiers in the ASE chain and a perfect parabolic gain profile centered around 1565 nm. The broadest obtainable pump bandwidth is shown with a dashed blue line. However, because the gain profiles of the amplifiers used in the experiment deviate somewhat from being purely parabolic and have slightly shifted gain peaks due to variations in doping concentration, fiber length, etc., the broadest bandwidth we could extract from our ASE-based pump system was $\sim 7\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$.

Although it is not possible to measure the full field of a CW source (as it is for, say, femtosecond-scale pulses using frequency resolved gating techniques, among others [36,37]), it is possible to obtain the average duration of the intensity fluctuations through AC of the field intensity, given by Eq. 10. Figure 7 shows the AC functions—both numerical and experimental—for three pump bandwidths: (a) and (b) 0.36 nm, (c) and (d) 1.77 nm, and (e) and (f) 5.28 nm. Comparison of the measured intensity AC with the calculated AC function of the simulated field provides a reliable means of evaluating the accuracy of our numerical model of the experimental pump system. Excellent agreement confirms that our model predicts empirically valid fluctuations in the pump field and can be used to generate the initial noise conditions for the simulation of SCG in a HNLF.

#### B. Modeling Supercontinuum Evolution in Fiber

Modeling of supercontinuum evolution in optical fiber waveguides can be well described by the one-dimensional GNLSE that includes the relevant linear and nonlinear contributions that modify the initial field depending on the parameters of the fiber, such as group velocity dispersion (GVD) and nonlinearity [23]. The frequency domain formulation we used in this work was [38]

In Eq. (11) the left-hand side terms model linear effects: the power attenuation $\alpha $, and the dispersive coefficients $\beta $, to arbitrary order. However here we consider only the GVD ${\beta}_{2}$ and third- and fourth-order dispersion ${\beta}^{(3,4)}$ terms of the Taylor series expansion. The right-hand side describes nonlinear effects, namely, the instantaneous contributions to the nonlinearity caused by the electronic Kerr effect, self-steepening, and optical shock formation; and the delayed contribution from noninstantaneous effects, namely, inelastic Raman scattering. The convolution integral contains an instantaneous electronic and a delayed Raman contribution, where the response function ${h}_{R}(t)$ can take a number of forms, depending on the complexity (and accuracy) of the desired function [23,39,40]; here we use a multivibrational-mode model developed by Hollenbeck and Cantrell [40].

When the duration of the temporal input field is of the order of hundreds of femtoseconds, the time scale of the simulation is typically performed over a reference frame several picoseconds long. This means that, for the case of a CW initial input, we simulate only a snapshot of the field in time, with a typical temporal grid width of the order of several hundred picoseconds—limited by the requirement to satisfy the Nyquist sampling theorem and obtain a suitable frequency coverage to contain the spectral expansion of the input field on propagation, usually resulting in the discretization of the temporal grid with approximately ${2}^{17}$ points.

For a detailed discussion of formulations of, and solutions to, the GNLSE in the context of modeling SCG in optical fibers, we refer to the recent monograph on supercontinuum by Taylor and Dudley [41], and references therein.

With CW continuum evolution seeded by noise driven processes,it is necessary to perform ensemble simulations, with different random initial pump conditions and subsequently average over the ensemble for good quantitative agreement between numerical and physical experiment [42].

## 5. RESULTS

The spectral output of the HNLF, under a fixed pump power of 6.3 W, for pump bandwidths in the range 0.3–7 nm, was recorded using an optical spectrum analyzer. The experimentally measured continuum width (10 dB) as a function of the pump source bandwidth (3 dB) is plotted in Fig. 8(a).

It is evident that, initially, the continuum width increases as the pump bandwidth increases. Beyond a pump bandwidth of $\sim 3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, the rate of increase in the corresponding continuum width slows and the spectral expansion begins to saturate. Indeed, beyond $\sim 5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, the spectral expansion appears to start to contract. Because of limitations in the pump system that have been discussed above, it is not possible to extend further the degree of pump source incoherence and recover the full contraction in the continuum evolution. However, with perfectly coincident gain profiles we showed that, with a numerical model of the pump system, we could obtain a maximum pump bandwidth of $\sim 11\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$. The dependence of the continuum width on the pump source bandwidth calculated using our numerical model is shown in Fig. 8(b). The additional degree of pump source incoherence (or pump bandwidth, up to $\sim 11\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$) shows a full saturation of the spectral expansion for pump bandwidths beyond 4 nm and a contraction above 6 nm. Each point on the curve in Fig. 8 represents an average over an ensemble of five simulations, each with a unique initial noise field. The peak of the curve is less well defined than demonstrated in the experimental measurement. This can be predominantly attributed to the fact that the ensemble size is too small—typically, CW SCG simulations will be averaged over $\sim 50$ single-shot simulations. However, the additional accuracy of simulating the initial noise field comes at increased computational cost.

Figure 9 shows the temporal field intensities for pump bandwidths of (a) 0.33, (c) 2.58, and (e) 6.24 nm, and the corresponding (single-shot) output spectrum [(b), (d), and (f), respectively] generated after 50 m of propagation in the HNLF; the input pump spectrum is shown with a dashed curve for reference. The calculated MI period for the HNLF fiber parameters is shown with the temporal field for comparison. When the pump source exhibits a long coherence time, the peak power enhancement is low and the corresponding output spectrum shows no continuum evolution after the full propagation length. As the coherence time of the initial field decreases and the peak power is enhanced, a broad continuum is formed after the full propagation length. As long as the pump coherence time is longer than the MI period, the intensity noise fluctuations enhance the soliton energies formed through MI and, hence, the resulting continuum evolution. Additionally, this spectrum shows the largest energy transfer to dispersive waves around 1300 nm [43,44], showing that the most intense and short duration solitons are formed in this case. The final panel of Fig. 9 shows that, when the intensity fluctuations in the initial time-domain field are much shorter than the MI period, the MI efficiency and the continuum expansion is reduced.

This dependence is further exemplified in Fig. 10, where the spectral intensity evolution is plotted against distance of propagation for three input pump bandwidths: (a) 0.56, (b) 4.25, and (c) 38.66 nm. Although the same ASE model was used to generate the initial pump fields for the SCG simulations, here we have relaxed the requirement on the components in the model to fully represent values of experimental parameters to allow us to generate an input field with a FWHM spectral bandwidth of 38.66 nm, and to show the full contraction of the continuum dynamics. The continuum dynamics are often better observed simultaneously in time and frequency space through the field spectrogram, where the temporal location of spectral components leads to an intuitive view of the nonlinear interaction within the fiber, and the evolution of soliton and dispersive processes. The field spectrograms (for the corresponding input pump bandwidths from Fig. 10) after 50 m of propagation of the initial field in the HNLF fiber are shown in Fig. 11. Solitonic structures can be identified as temporally and spectrally localized hot spots, while dispersive waves exhibit lower intensity and a temporal chirp. Figure 11(b), with a pump bandwidth of 4.25 nm, shows the only significant continuum formation, with the generation and redshifting of solitons to the long-wavelength edge of the pump wavelength and dispersive wave radiation to the short edge (albeit $>45\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ down from the peak of the pump) indicating the formation of broadband (i.e., ultrashort) solitons. As the red part of the spectrum extends, this dispersive radiation blueshifts, indicating the presence of soliton trapped dispersive waves [22,45]. In contrast, no solitons are formed in Fig. 11(a), where the input pump bandwidth is 0.56 nm, although the pump has experienced a small degree of self-phase modulation over the full propagation length, seen in the chirping of the temporal intensity peaks. As the pump bandwidth becomes too broad and the frequency of temporal fluctuations in the time-domain field far exceed the MI period, MI is effectively quenched and the pump wave does not efficiently break down into a train of fundamental solitons [Fig. 11(c)].

## 6. DISCUSSION

The modeling of the pump source in the previous sections, which provided the initial conditions for the supercontinuum simulations, was related as closely as possible to the experimental system, to allow direct comparison between numerical and experimental results. In this section we consider a simplified model of the pump system to generate the initial pump conditions, but use the same propagation equations to simulate the evolution of the field in the HNLF. The simplified model allows a broader space of parameters to be investigated.

It has been shown that a CW-fiber laser can be well modeled by a sech-shaped spectral intensity profile with an associated random spectral phase to account for intensity fluctuations in the time-domain field [13], in a manner similar to that described in [26]. This simple model allows arbitrarily broad bandwidth pump sources to be modeled by modification to the width of the sech-shaped spectrum. Here, we use this model to provide the initial conditions for simulations performed with only first-order dispersion (${\beta}_{2}$) and no high-order dispersion effects. Although this model exhibits an unphysical dependence on the numerical grid size and ignores any phase relationship between laser modes, it is self-consistent for simulations conducted over constant grids and does not require extensive numerical evaluation.

To probe the dependence of the MI/continuum efficiency on pump bandwidth, we study a range of fibers designed to have a range of MI periods. We do this by varying ${\beta}_{2}$ while holding $\gamma =44\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{W}}^{-1}\text{\hspace{0.17em}}{\mathrm{km}}^{-1}$ constant. This means that the MI gain and the nonlinear length of the fiber is fixed. In the following, we choose an average pump power of 10 W and a fiber length of 20 m. We illustrate the results on a wavelength scale centered at 1065 nm, similar to Yb fiber lasers (and the fiber parameters are similar to those used in common CW continuum experiments at 1065 nm in photonic crystal fibers), but the results are generally applicable.

Figure 12 shows the output spectra for pumping a fiber with ${\beta}_{2}=-0.012\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{ps}}^{2}\text{\hspace{0.17em}}{\mathrm{km}}^{-1}$ with a range of pump bandwidths. We see that there is a clear optimum pump bandwidth of $\sim 2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ to obtain the broadest supercontinuum. This corresponds to a coherence time of $\sim 1.7\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{ps}$, compared to an MI period of 0.74 ps.

Figure 13(a) shows obtained supercontinuum width (20 dB level) as a function of a 3 dB pump bandwidth for a number of fibers with differing values of ${\beta}_{2}$ designed to show a linear progression in the quantity $\sqrt{\gamma /|{\beta}_{2}|}$, which, for fixed pump power, determines the MI bandwidth and, hence, the MI period. Each curve consists of 20 pump bandwidths logarithmically distributed between 0.03 and 30 nm, where each point is averaged over 40 single-shot simulations with different random initial noise conditions. It is clear that, for each curve, there is a variation in obtained supercontinuum width as a function of pump bandwidth, with a gentle peak value.

As described in Section 2, in the absence of higher-order dispersion, optimal continuum formation is achieved in a fiber with a high nonlinearity and low value of GVD; consequently, a large value of $\sqrt{\gamma /|{\beta}_{2}|}$ is desirable. This is evident in Fig. 13(a), as the continuum bandwidth clearly scales with this quantity. The reason for this scaling is that the MI period and, hence, the induced soliton duration, is reduced. As the MI period is reduced, the tolerated pump incoherence is higher and, hence, the range of pump bandwidths over which a continuum will form should increase with increasing $\sqrt{\gamma /|{\beta}_{2}|}$; this is also clear from Fig. 13(a): the curves are wider for increasing $\sqrt{\gamma /|{\beta}_{2}|}$. The peak value in these curves, i.e., the pump bandwidth leading to the broadest continuum, also depends on $\sqrt{\gamma /|{\beta}_{2}|}$, as expected from the preceding analysis. This allows us to express an optimum pump bandwidth as a function of the MI bandwidth; the relationship, plotted in Fig. 13(b), is found to be linear where the optimum pump bandwidth is approximately one-third of the MI bandwidth: $\mathrm{\Delta}{\omega}_{\text{pump}}\approx 0.3\mathrm{\Delta}{\omega}_{\mathrm{MI}}$.

Taking the coherence time of the ${\mathrm{sech}}^{2}$ spectrum as $5.53/(\mathrm{\Delta}{\omega}_{\text{pump}})$ [21], this implies that the optimum pump coherence time ${\tau}_{\text{pump}}\approx 3{T}_{\mathrm{MI}}$. This agrees with our original hypothesis: the optimum degree of coherence of the pump source is the minimum (enhancing peak power fluctuations) that remains sufficiently coherent for MI to occur.

## 7. CONCLUSION

In conclusion, we have studied in detail, both experimentally and numerically, the relationship between the degree of pump coherence and the generation and evolution of a CW supercontinuum from MI. We have developed a model to accurately reproduce the temporal noise field of a CW laser; direct measurements of the intensity fluctuations through the intensity AC confirm the validity of our numerical approach. This CW model is used to create the initial conditions for the simulation of CW-pumped supercontinuum in HNLFs. While it is well known that wave incoherence destroys MI, we have shown empirically that partial wave coherence of the pump laser can enhance the instantaneous peak power, leading to the generation of shorter and more intense solitons through MI. This results in the formation of a broad spanning supercontinuum that cannot be generated with a coherent input matching all other parameters. Furthermore, we have shown that an optimal degree of pump coherence exists that results in the greatest amount of spectral broadening of the initial pump spectrum. The optimum pump bandwidth is that which supports the largest possible intensity fluctuations, while remaining sufficiently coherent for MI to occur. It is found that the optimum pump bandwidth shifts linearly with the MI frequency, such that we can define an optimum bandwidth as approximately one-third of the MI frequency. Current theory of incoherent MI does not consider the enhancement of the peak power due to partial coherence of the input wave, and requires augmentation to account for the empirical observations presented in this paper.

## ACKNOWLEDGMENTS

E. J. R. Kelleher is supported by a studentship from the Engineering and Physical Sciences Research Council (EPSRC).

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