We numerically and experimentally demonstrate efficient generation of an equalized optical comb with 150-nm bandwidth. The comb was generated by low-power, continuous-wave seeds, eliminating the need for pulsed laser sources. The new architecture relies on efficient creation of higher-order mixing tones in phase-matched nonlinear fiber stages separated by a linear compressor. Wideband generation was enabled by precise dispersion engineering of multiple-stage parametric mixers.
©2012 Optical Society of America
A wideband generation of frequency combs in the optical domain has been demonstrated in both crystalline and fiber devices [1–11], spanning the spectral range from visible to infrared bands. Since early demonstrations, compact comb devices have been of particular interest in remote sensing , spectroscopy , clock distribution , ranging , photonics-to-RF frequency mapping , and waveform synthesis . As a result, current efforts remain focused on increasing generation efficiency and decreasing physical size [3,4], without compromising the comb bandwidth or noise performance. This trend is expected to lead to ubiquitous use of frequency combs, away from specialized applications that can tolerate instruments with high complexity and dissipation levels.
Common comb generation architecture relies on mode-locked lasers (MLL)  to seed the nonlinear process and define spectral tone distribution. Coupled with a feedback mechanism , this approach has led to demonstrations of wideband and low-noise devices used in wave-forming, ranging and spectroscopy.
In an alternative to the conventional approach, a frequency comb synthesis can be accomplished without a MLL seed. Originally proposed for optical communication , this technique relies on four-photon mixing (FPM), seeded by the modulated optical carrier. The method eliminates the mode-locked seed thus decoupling the comb frequency pitch and MLL cavity properties. However, the cavity-less generation also imposes severe phase matching requirements on the parametric mixer used to generate multiple frequency tones. Indeed, it was shown earlier  that precise mixer dispersion management is critical as it defines bandwidth, efficiency and noise properties of the parametric response.
Recognizing this challenge, we investigate and report on the performance of new mixer design and its application to the generation of broadband frequency combs. The device is seeded by two CW tones that can be freely tuned to generate an arbitrary comb frequency pitch, in contrast to the cavity-defined approach. The new multistage mixer incorporates strain-synthesized dispersion to achieve precise phase matching necessary for efficient wideband frequency generation.
While multistage, synthetic-dispersion mixer can be operated with different frequency stabilization and noise reduction techniques, this report specifically focuses on power-efficient design only; a subsequent report will describe the noise-management and stabilization aspects. The paper is organized as follows: the second section introduces the notion of multistage mixer; the third section describes requirements imposed by dispersion engineering; the experimental demonstration is reported in the last section.
2. CW-Seeded frequency comb generation in a multi-stage parametric mixer
In contrast to modulator-driven mixer , seeded by three frequency tones, we consider comb generation initiated by a two-tone mixing process. In the simplest topology, two frequency non-degenerate waves propagating along the nonlinear waveguide will act as parametric pumps  and will create new spectral sidebands. With sufficient launch power, newly created tones will grow and contribute to FPM process, generating additional, high-order frequency tones . In case when a parametric mixer is engineered to maintain phase synchronization among the interacting waves, the two-pump seeding will lead to a high-order tone generation across its entire operating bandwidth. In practical terms, the efficiency of such device depends on the pump power and the degree of phase matching that can be achieved across the operating bandwidth.
Even in the case of ideal phase matching, the FPM efficiency is a direct function of the mixer figure of merit (FoM), defined as the product of waveguide nonlinearity (γ), pump power (P) and the effective interaction length (L) . To maximize the FoM, it is tempting to use high pump power; such choice would also provide the means for shorter, uniform mixers while maintaining high FoM. Unfortunately, the use of intense CW pump in fibers necessarily leads to broadband noise generation via Raman scattering and parametric vacuum noise amplification . More importantly, Brillouin scattering sets the maximal CW pump level to only tens of milli-Watts in highly nonlinear fibers (HNLF), of primary interest in this work . An attempt to increase the Brillouin threshold by pump dithering also eliminates any possibility of narrow-tone comb generation . Even if Brillouin limit in fiber did not exist, the requirement for a high-power CW seed would drastically reduce the practicality of any comb generation scheme. Consequently, the optimal parametric mixer design should be seeded by a CW power compatible with conventional (low power) diode pumps, and be capable of wideband frequency generation.
The requirement for a high power CW seed can be removed in a multiple-stage mixer design  by recognizing that the cumulative FoM can be increased by enhancing the intensity within the mixer waveguide. The simplest form of field enhancement can be accomplished by compressing the two-tone seed beat waveform in the temporal domain, as illustrated in Fig. 1 . The first nonlinear stage serves as a nonlinear phase-induction element: a tone-tone beat experiences a Kerr-induced positive chirp and leads to a limited generation of higher order tones. In the next step, the chirped waveform is compressed in a short segment of single-mode fiber (SMF) possessing negative second-order dispersion (β2). Subsequently, the compressed waveform enters the second nonlinear stage with higher peak power, allowing the second mixer to operate with a correspondingly larger FoM. Consequently, FPM interaction in a second nonlinear stage results in a high-order tone count, as required for wideband frequency comb generation. This chirp-compress-and-mix cycle can be repeated multiple times, allowing one, at least in principle, to reach very high FoMs in the concluding mixer stage and to create very wide band response.
The implementation of the staged frequency generation scheme requires precise dispersion engineering leading to the optimal compression in the last mixer section. Consequently, the goal of the first stage is to initiate the multiple-tone generation and induce chirp before entering the compressor. The Kerr-effect origin of the aforementioned chirping implies that the compression ratio, and correspondingly the peak power, is proportional to the number of tones. In fact, the compression ration can be expressed in simple terms if one neglects higher-order dispersion. If the first stage is seeded by two CW tones characterized by the average power P and separated by an angular frequency interval Δω, the beat intensity will be given by:
The joint two-tone propagation along a fiber section of length L and possessing a nonlinear coefficient of γ will induce a time dependent nonlinear phase shift of ϕ(t) = γP(t)L. In analogy with chirped Gaussian pulse analysis , let us introduce a dimensionless chirp parameter CEq. (1), the characteristic time t0 isEq. (3) and Eq. (1), the corresponding chirp parameter is
It is clearly seen from Eq. (4) that the chirp parameter contains higher order terms, however, within the beat time interval (i.e. the period) (~2π/Δω), as a first-order approximation, one can consider compensating only the quadratic part of the phase shift ϕ(t) and in the vicinity of the pulse peak:
For a Gaussian waveform, the optimum length of the compressor LC is defined by :
Let us consider a practical example in which two CW tones (pumps) with 1 W power are launched into a 100-m long fiber with a nonlinear coefficient of 20 /W/km. Equation (5) then defines the corresponding chirp parameter C to be 13.5. If pumps waves are separated by 1 nm, the characteristic time is t0 = η/2Δf = 2.35 ps. For such a pulse duration, the characteristic dispersion length LD in a conventional single-mode fiber (SMF) with β2 = 2·10−26 s2/m is 270 m and the optimum length of the compressor is LC = 20 m (Eq. (6)).
On the other hand, note that according to Eq. (5), the effective chirp generation is governed by the HNLF type and its confinement factor (or, equivalently the nonlinearity γ). In practice, it is generally preferable to use fiber with a high nonlinear coefficient γ (> 20 /W/km), rather than longer fiber interaction length (L). The length of the first stage should be short enough in order to avoid pump back-reflection due to stimulated Brillouin scattering (SBS) and to minimize any polarization-mode dispersion impairments. While the SBS can be suppressed by applying longitudinally varying tension , this method typically leads to Brillouin threshold increase of 15 dB in standard 100-m HNLF segment before the onset of ancillary impairments. Besides reduction in interaction length, the high-gamma choice is equivalent to higher confinement waveguides, which, in turn, requires smaller stretching force, allowing for higher tension gradient to be induced below the mechanical breaking limit.
In addition to its length and nonlinearity, the choice of zero dispersion wavelength (ZDW) of the first (chirp-inducing) stage is constrained by multiple considerations. Firstly, the fiber dispersion at the pump (seed) frequencies should be low enough to guarantee phase matching. Consequently, both seed wavelengths are expected to be placed in the vicinity of the ZDW. Secondly, in order to suppress the parametric noise induced by the modulation instability, pump wavelengths should be placed within the normal dispersion band. If the slope of the fiber is low enough (< 0.03 ps/nm2/km), the position of the ZDW can be varied and does not require accuracy greater than few nanometers. As a result, ZDW variation induced by the longitudinal tensioning does not degrade the seed-seed beat efficiency. Finally, it is important to emphasize that the generation of multiple frequency tones within the first stage, while not the primary goal of the design, also increases the SBS threshold as the power distribution of the total field now occupies multiple comb lines.
The second (compressor) stage, if designed properly, forms a high-peak waveform in temporal domain, enabling the last stage to operate as a high FoM mixer. The third stage is also subject to the strictest dispersion control requirements: while the first stage mixer generates only few pump-pump mixing tones, the third stage is expected to generate spectrally flat, wideband frequency plan accommodating hundreds of tones. This means that the intra-mixer dispersion must be sufficiently low across the entire generating band of interest. In practical terms, this goal can be accomplished by a dispersion flattened HNLF (DF-HNLF) possessing a parabolic dispersion profile. The goal of this work was to generate a frequency comb within a 200 nm window, requiring that the absolute value of the HNLF dispersion does not exceed 1 ps/nm/km within this range. Unfortunately, while the dispersive properties of the latest generation of DF-HNLF  satisfy the simplified dispersive requirement, it can also be shown that an attempt to generate CW-seeded frequency comb generally fails, as illustrated in Fig. 2 . The following section describes the underlying failure mechanism that prevents a wideband comb generation and outlines a procedure for its removal.
3. Absolute dispersion accuracy and its role in frequency comb generation
The measured frequency combs, shown in Fig. 2, indicate that small variations along the DF-HNLF can lead to qualitatively different device response. To investigate these, we have constructed model of the staged comb generator, as illustrated in Fig. 3(a) . The model uses a rigorous nonlinear Schrödinger solver with experimentally measured physical characteristics of both fiber types and includes vacuum noise seeding to accurately predict the optical signal-to-noise ratio (OSNR) of the newly generated tones. In all simulations, the amplified spontaneous noise of the amplifies A1 and A2 was not considered.
In the first set of simulations, the first (chirping) mixer (HNLF1) had a nonlinear coefficient of 12.9 /W/km (n2 = 3.5⋅10−20 m2/W, Aeff = 11 um2), an attenuation of 0.8 dB/km, a dispersion slope of 0.025ps/nm2/km, and a fixed length of 105 m; the pump seed wavelengths were fixed. The ZDW of the stage was set to 1540 and 1550 nm, referring anomalous and normal dispersive regimes for the interacting pumps. The third stage (HNLF2) was 200-m long, had a nonlinear coefficient of 9.5 /W/km (n2 = 3.5⋅10−20 m2/W, Aeff = 15 um2), an attenuation of 0.5 dB/km, and dispersion modeled after measured profile (Fig. 3(b)). As noted earlier, the dispersion profile guaranteed that both pump seeds operate in the normal dispersion regime (D < 0), with maximal dispersion of −0.025 ps/nm2/km occurring at 1546 nm. The pump-pump frequency pitch was varied from 0.5 nm in Fig. 4 , to 1.0 nm in Fig. 5 and finally to 2 nm in Fig. 6 . Each case was also considered with low (P = 0.5W) and high power (P = 1.0 W) pump launch conditions. The length of the compressor stage (SMF1) was optimized for each pump launch condition, as stated in captions of Figs. 4-6.
All simulated examples clearly demonstrate that the positioning of the pump waves within the anomalous band of the HNLF1 fiber causes significant OSNR degradation for the generated comb lines. For example, for 1-W pump power and 1-nm pump spacing, OSNR degrades by more than 10 dB relative to the case when the pumps are placed within the normal dispersion regime. When pump separation is halved to 0.5 nm and pump power remains 1 W, a 100-nm wide comb can easily be generated. In this case, strong local coupling between closely spaced tones also results in significant modulation of FPM interaction across the spectrum (Fig. 4). By increasing the seeding tones′ separation, the comb power profile becomes equalized, accompanied by an aggregate bandwidth increase. Indeed, when the pump-pump pitch is doubled to 2 nm, a 150-nm wide, power-equalized comb can be generated (Fig. 6). Finally, separate simulations, not shown here, investigated the performance of a single, dispersion tailored generator stage. These indicate that generation of a 100-nm wide comb using 500-mW pump limit is not feasible with currently available DF-HNLFs. Finally, we note that the simulations were done for CW pumps that did not include the pump frequency noise. In the case when the pump laser has finite linewidth, the frequency noise amplified by modulation instability provides additional degradation of OSNR and a progressive spectral broadening of the comb lines .
To further increase the comb bandwidth in the case when low-power (500-mW) pumps are used, the generator architecture is modified by adding a compressor and a mixer stage, as illustrated in Fig. 7 .
The new topology uses diode-power rated 500 mW pumps separated by 1 nm. The second (HNLF2) and the third (HNLF3) mixer stages were 100 and 150m long, respectively, while the newly optimized lengths of the compressors SMF1 and SMF2 were 64 and 6.5 m. The front chirping block was retained and was identical to the unit used earlier with ZDW of 1550 nm, whereas the absolute value of the dispersive maxima of the parabolic profile (Fig. 3(b)) was varied to investigate its impact on the frequency comb generation and is plotted in Figs. 8 -9 .
When the dispersive peak of both DF-HNLFs (HNLF2 and HNLF3) is well above zero (DPeak > 0.4ps/nm/km), a comb spectral profile has an approximately triangular shape and is characterized by a low OSNR that varies between 25 and 30 dB, when measured in a 0.1 nm reference bandwidth. In case when the dispersive peak becomes weakly positive (0 < DPeak < 0.2 ps/nm/km) multistage FPM interaction leads to a significant enhancement of the parametric noise and the modulation instability dominates the frequency comb generation. In the temporal domain, this parameter set is manifested by disintegration of the original pump-pump beat pattern into a sequence of intermittent, randomly scattered, short pulses.
In sharp contrast, in case when the maximal DF-HNLFs’ dispersion remains within a narrow dispersion-negative band (−0.1 ps/nm/km < DPeak < 0), a regular wideband FPM cascade is accompanied by simultaneous suppression of the parametric fluorescence. As a result, the OSNR of the newly generated tones is high and exceeds 40 dB. A further decrease (i.e. absolute increase) of the peak dispersion affects the comb bandwidth while maintaining the OSNR. In summary, this implies that fabrication tolerance of 0.1 ps/nm/km is required in order to attain combs wider than 100 nm using the parabolic dispersion HNLF profile. Unfortunately, the last requirement also means that the core fluctuation of such fiber must not exceed 0.1% (approximately 1 nm). This last fact clearly explains widely varied response obtained from the single fiber draw (Fig. 2). Nevertheless, the finding also indicates a clear path that needs to be taken to synthesize wider-than 100 nm, high-OSNR frequency comb. Indeed, the bandwidth of the comb generated by inclusion of an additional stage (SMF2 + HNLF3, Fig. 7) is almost tripled (Fig. 9(a) versus Fig. 5(a)) in comparison to an original scheme (Fig. 3(a)), regardless of the additional splicing loss. The following section describes the construction of such a device.
4. Experimental results: strain-synthesized wideband frequency comb
The modeling study described in the previous section revealed that the DF-HNLF-based mixer dispersion must be controlled with accuracy better than 0.1 ps/nm/km. Rather than insisting on impractical fabrication of such a fiber, we recognize that longitudinal straining of HNLF  can be used to synthesize the required dispersion target.
Consequently, an experimental setup containing multiple stages was constructed according to the topology shown in Fig. 7. Two lasers with wavelengths of λP1 = 1552.30 and λP2 = 1553.36 nm possessing sub-10 kHz linewidths were amplified by erbium-doped fiber amplifiers (EDFA). The amplified spontaneous noise was removed by pair of cascaded broad (0.6-nm-wide) filters. Pumps were combined using a low-loss combiner (WDM) and launched into the first 105-m-long nonlinear fiber (HNLF1) possessing a nonlinear coefficient of 19.6 /km/W, a dispersion slope of 0.026 ps/nm2/km, and a global ZDW of 1562 nm. This fiber was divided into 10 sections and stretched by applying a specific tension plan. The stretching map was devised to guarantee that the consecutive HNLF sections experience the highest strain difference while the average tension was kept approximately constant across the entire fiber length . Correspondingly, the SBS threshold was made to exceed one Watt of launched power. The pump powers before HNLF1 were set to 27 dBm. Subsequently, the chirped pump-pump beat was compressed in a 59-m long single mode fiber (SMF1) and sent to a 130-m-long DF-HNLF (HNLF2) with a nonlinear coefficient of 12 /W/km. The HNLF2 had a maximum dispersion below zero so that dispersion correction was not required. The second compression was performed by a 6-m long SMF fiber (SMF2) and final mixing was accomplished in a 240-m long DF-HNLF (HNLF3) with physical properties identical to HNLF2. The polarization states of propagated signals were aligned before entering nonlinear stages to maximize the nonlinear efficiency.
The group delays of the 240-m-long HNLF3 fiber were measured multiple times in order to obtain accurate strained and unstrained dispersion profiles, as shown in Fig. 10(a) . The group delays were fitted by the 4th order Sellmeier expression (solid lines in Fig. 10(a)) to obtain the corresponding dispersion profiles (Fig. 10(b)). In contrast to HNLF2, the HNLF3 possessed a DPeak of 0.3 ps/nm/km before straining, well in excess of what is identified as the optimum, leading to the expectation of low-OSNR comb generation. However, the straining process has succeeded in reducing the peak value of HNLF3 dispersion to near-zero which, as explained above, resulted in a significant qualitative improvement of the generated comb.
The performance of the dispersive correction is tested by measuring the spectral response of the multistage comb generator. In the first measurement, the frequency comb corresponding to the intrinsic (i.e. unstrained) HNLF3 section is plotted in Fig. 11 and illustrates low-OSNR tone generation that is almost indistinguishable from the parametric noise, as predicted from simulation (Figs. 8(b)-8(c)). The measurement verifies the finding from the previous section: low positive DF-HNLF peak dispersion is the worst case choice for the mixer stage. As a second measurement, the HNLF3 section was dispersion corrected by applying a longitudinally-constant force. In sharp contrast to the first measurement, Fig. 12 plots an equalized, wider-than-100 nm comb possessing minimum OSNR in excess of 35 dB. It is also important to emphasize qualitative similarity between simulated and measured spectra for both original (Figs. 8(b)-8(c) and Fig. 11) and corrected dispersion (Fig. 9(a) and Fig. 12).
We reported simulated and experimental results for a multi-stage all-fiber-based parametric mixer seeded by 500-mW pumps. The scheme is capable of generating spectrally flat wideband CW comb with a signal-to-noise ratio higher than 30 dB. Reported mixer performance has been achieved by precise dispersion management of all linear and nonlinear stages. The wideband comb generation was achieved by using tension-engineered dispersion flattened highly nonlinear fibers. Finally, we demonstrated a practical way for precise tuning of the dispersion profile of DF-HNLF fibers which results in qualitative improvement in parametric mixer response.
This work was supported by Defense Advanced Research Project Agency's Microtechnology Office. We gratefully acknowledge Sumitomo Electric Industries for providing the fiber samples used in this work.
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