Abstract

We demonstrate new technique for generation of programmable-pitch, wideband frequency combs with low phase noise. The comb generation was achieved using cavity-less, multistage mixer driven by two tunable continuous-wave pump seeds. The approach relies on phase-correlated continuous-wave pumps in order to cancel spectral linewidth broadening inherent to parametric comb generation. Parametric combs with over 200-nm bandwidth were obtained and characterized with respect to phase noise scaling to demonstrate linewidth preservation over 100 generated tones.

©2012 Optical Society of America

1. Introduction

Optical frequency combs (OFC) used as an optical reference has increased metrology accuracy [13] and qualitatively redefined practicality of wide range of applications [4]. These include high-resolution spectroscopy [5], optical referencing of atomic clocks [6], arbitrary optical waveform generation [7], low-noise microwave synthesis [8], high-capacity optical coherent communications [9] and astronomical spectrograph calibration [10]. OFC generation typically relies on mode-locked lasers [3] or precisely stabilized optical resonators [11]. These approaches can also include nonlinear interaction to aid spectral broadening, allowing frequency generation over an octave to map an optical frequency to microwave domain in a self-referenced manner [12]. Both approaches critically depend on optical cavity structure that inherently defines frequency stability and prevents the change in comb frequency pitch.

It is possible, at least in principle, to decouple an efficient OFC generation from cavity imposed limits. Indeed, an OFC can be practically realized in a cavity-less manner, using either electro-optic modulators [13] or travelling-wave four-wave mixing (FWM) [14,15]. While the former offers a simple mean for OFC creation, its bandwidth is generally lower than that of FWM generated comb due to finite electrical bandwidth of electro-optical modulator device. In contrast, phase-matched FWM process is capable of generating a wideband comb while preserving phase reference (lock) among all generated frequency tones. More importantly, travelling-wave (cavity-less) generation offers intrinsic flexibility in regards to tone spacing, allowing for nearly arbitrary frequency pitch. FWM-generated (parametric) comb frequency pitch is defined by pump-pump separation that can be freely adjusted to meet frequency plans posing a challenge for mode-locked comb generation.

In practice, the frequency tunability must also be accompanied by power efficient, spectrally equalized FWM process. In the simplest scheme, dual continuous-wave (CW) pump seeding was used to generate OFCs with bandwidths exceeding 200 nm in dispersion-synthesized multistage mixers [16]. A tunable-pitch parametric OFC seeded by free-running pumps has been recently reported [17] with spectral flatness of less than 10 dB and optical signal-to-noise ratio (OSNR) of more than 35 dB, indicating its practical potential. In addition, cavity-less OFC generation also provides for unique signal replication capability that has no equivalent with conventional OFC techniques. By injecting an optical signal along with two comb-seeding pumps, it is possible to replicate the optical field across the entire OFC bandwidth [16,18]. This feature has enabled unique signal processing techniques in both analog and digital domain [19].

Unfortunately, cavity-less parametric comb generation is also severely impaired by the lack of any obvious mechanism inhibiting noise and spectral linewidth growth [16]. Indeed, parametric generation leads to progressive growth of the spectral tone linewidth that scales linearly or quadratically with the FWM order [16]. Although this effect may be instrumental in other applications [20,21], it significantly degrades the quality of the higher-order frequency tones and directly contributes to OSNR decrease in comb generation. The coherence degradation of higher-order tones inevitably limits the applicability of parametric cavity-less combs in all applications that rely on high phase fidelity [22]. The impairment mechanism was recognized early [23], and originated with the fact that two pump seeds were represented by free-running lasers. As a consequence, two FWM seed waves were phase-uncorrelated, leading to uninhibited linewidth and noise growth.

Recognizing this basic limitation, we have constructed parametric comb generator seeded by two phase-correlated CW pumps and investigated its performance. The phase correlation was achieved by injection locking to the phase-modulated master source to achieve true pump-pump coherence mapping. The new OFC possessed bandwidth in excess of 200 nm and was measured to have dramatic suppression of higher-order-tone spectral linewidth broadening. Efficient comb generation required pump seeding well above Brillouin threshold of the highly nonlinear fiber (HNLF). Strict preservation and characterization of higher-order tone coherency prohibited conventional Brillouin suppression scheme based on pump dithering [24,25]. Consequently, the injection-locked pumps were used to seed multistage mixer incorporating strain-induced Brillouin suppression [26]. The new mixer capable of highly coherent high-order tone generation, opens a practical path to continuously reconfigurable, CW-driven coherent comb devices.

The reminder of the paper is organized as follows: in Section 2, the linewidth characteristics of parametric combs are described, and the pump phase correlation principle is introduced; Section 3 describes the experimental architecture; in Section 4 we compare the measured performance of combs generated by phase-correlated and free-running pump seeds. Section 4 also illustrates unique features of a cavity-less, pump-correlated OFC such as variable frequency pitch. Finally, the impairment mechanisms leading to residual uncorrelated phase noise and their suppression are discussed in Section 5.

2. Self-seeded, phase-correlated parametric generation

A two-pump seeded parametric comb generation can be understood as a cascade of degenerate FWM interactions in which signal-pump-idler mixing defines successively higher-order tone generation. The mixing process is initiated by creation of the first-order tone possessing the phase satisfying the well-known relationship [23,27]:

θa1,s1=2θP1,P2θP2,P1
where subscripts a1,s1 represent the first-order anti-Stokes (high frequency) and Stokes (low frequency) lines, respectively;P1,P2 denote two pump waves as marked in Fig. 1(a) . The cascade of FWM described by Eq. (1) then leads to the scaling of phase in the higher-order FWM components as follows [28]:
θaN,sN=(N+1)θP1,P2NθP2,P1,
where N represents the N-th order comb line.

 figure: Fig. 1

Fig. 1 Illustration of linewidth scaling in a two-pump seeded parametric comb with (a) un-correlated and (b) correlated pump phases.

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While the phase-noise scaling law is invariant to the lasers’ phase-noise statistics, the manifestation of the scaling in terms of line-shape and width of an individual tone is vastly affected by the spectral distribution of the laser frequency noise, and therefore, can only be revealed if the laser noise characteristics are known [29]. In one regime where the pump tones are perturbed only by an achromatic (white) frequency noise, the line-shapes of the tones will follow the Lorentzian profiles with their full-width-at-half-maximum (FWHM) linewidths (denoted as linewidth hereinafter) proportional to the squared phase noise amplitude. By taking a common assumption that the noise of the pump lasers are uncorrelated, the linewidth of a higher-order FWM component is found to scale with its order number in accordance to the following relationship [23,29]:

δvaN,sN=(N+1)2δvP1,P2+N2δvP2,P1,
where δv is the linewidth. As a result, under the assumption of white Gaussian frequency noise, the linewidth of a parametric comb seeded by two uncorrelated pumps scales quadratically with the order number.

On the other hand, the line-shape of a frequency tone is well approximated by a Gaussian shape if the underlying frequency noise is dominated by an 1/f-type spectral distribution (i.e. the noise power spectral density is inversely proportional to the frequency) [29,30]. In this scenario, the linewidth is directly proportional to the order number:

δvaN,sN=(N+1)δvP1,P2+NδvP2,P1,
in a FWM-generated comb. In practice, both white and 1/f frequency noise components are present in optical oscillators, resulting in a measured comb-line scaling function that falls between linear and quadratic dependency.

Although Eq. (2) depicts that the linewidth of a FWM product is inevitably broadened in configurations involving independent lasers, it also implies that the linewidth broadening can be eliminated, at least in principle, by correlating the seed pumps' phases:

θP1=θP2+θ0,
where θ0 is a static phase offset. In this case Eq. (2) reduces to
θaN,sN=θP1,P2±Nθ0,
where the constant term Nθ0 can be omitted in estimating the phase noise imposed on higher-order frequency terms. Equation (5) clearly indicates that phase characteristics of each comb line will be strictly preserved when using a phase-correlated pump pair, as shown in Fig. 1(b), rather than being multiplicatively broadened when pumps are free-running lasers.

The simplest method of generating phase-correlated optical waves at distinct, albeit closely spaced frequencies relies on phase modulation of a single laser carrier. This approach, however, is often characterized by significant decrease in OSNR of newly generated tones. Consequently, the phase modulation approach weighed against practically available OSNR of the seed, since an excess pump noise leads to progressively more noisy parametric generation [31]. Optical injection locking, in additional to ensuring phase-correlation between two distinct sources, can also address significant loss of OSNR prior to the mixing process. In practical terms, the injection locking serves the purpose of a regenerative amplifier, capable of recovering substantial decrease in OSNR, while at the same time guaranteeing strict phase relation between the frequency tones generated via phase modulation [32]. We note that similar techniques have been reported recently and have been successfully implemented in phase-sensitive amplification [3335] and high-repetition-rate pulse generation [36].

3. Experimental setup

Consequently, the experimental architecture was constructed, as shown in Fig. 2 . Two cavity-less parametric OFCs were generated with correlated (free-running) and uncorrelated pumps, Fig. 2(a) and 2(b), respectively. In Fig. 2(a), a narrow-linewidth (< 5-kHz) external-cavity-laser (ECL) was used as the master laser centered at 1549.3 nm. It was followed by two concatenated phase modulators generating an optical frequency comb spanning 5 nm with 25-GHz pitch, as shown in the inset. Next, two comb lines with 400-GHz (1547.7 and 1550.9 nm) or 200-GHz (1548.5 and 1550.1 nm) spacing were selected by a programmable optical bandpass fitler, and were used to injection lock two distributed-feedback (DFB) slave lasers characterized by 700-kHz linewidths. It should be noted that this method results in creation of two phase-correlated pumps with no loss in original laser OSNR, but these pumps can be generated at nearly arbitrary frequency spacing, limited only by one's ability to generate sufficiently wide phase-modulated spectrum. In practice, this means that combs possessing pitches in excess of 500 GHz are readily realized (the lower limit is only dictated by the narrowband filtering capability to separate and re-combine the two pump lines, which currently allows about GHz pump spacing). The flexibility of the cavity-less arrangement also overcomes one of the most important limitations associated with conventional mode-locked-laser based combs that must overcome significant challenge in achieving the repetition-rate higher than 10-GHz [4].

 figure: Fig. 2

Fig. 2 Experimental setup of parametric comb generation seeded by (a) two injection locked CW lasers with fully correlated phase relations and 200 GHz and 400 GHz frequency pitches, and (b) two free-running seed injection locked CW lasers with un-correlated phase relation and 400 GHz frequency pitch. ML: master laser, SL: slave laser, PM: phase modulator, PS: phase shifter, POBF: programmable optical bandpass filter, PC: polarization controller, BPF: bandpass filter, WDM: wavelength division multiplexer.

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The example phase noise spectra of the free-running distributed-feedback (DFB), external cavity laser (ECL) and injection-locked DFB lasers are shown in Fig. 3(a) at −10-dBm injection power that corresponds to a −30-dB injection ratio at 20-dBm slave laser output [37]. The spectra in Fig. 3 clearly shows that the slave laser phase noise strictly follows that of the master laser. Equally important, the amplitude noise is also dictated by the slave laser possessing modified relaxation oscillation peak [3335], as shown in Fig. 3(b). Consequently, both low phase and amplitude noise can be simultaneously obtained if proper matching between slave and master laser devices is made. Indeed, after the injection locking, the slave laser OSNR was measured to be 62-dB at 20-dBm output power, guaranteeing superior noise performance in subsequent parametric mixing stage.

 figure: Fig. 3

Fig. 3 Comparison of characterized (a) linewidths and (b) RIN spectra of the master ECL, free-running DFB laser and injection locked DFB laser, respectively.

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Subsequently, the output of each slave laser was amplified by an erbium doped fiber amplifier (EDFA) to 600 mW and then filtered, combined with the second pump seed and finally launched into a 3-stage highly-nonlinear-fiber (HNLF) multistage mixer [16,17]. The first stage of the mixer was constructed using a 105-m long HNLF with 1554-nm average zero dispersion wavelength, 0.021-ps/nm2/km dispersion slope and 22-W−1km−1 nonlinear coefficient. This section (HNLF1) was longitudinally strained to increase the Brillouin threshold [26] beyond that of CW pump level. The second, compression stage was composed of 4-m long standard single-mode-fiber (SMF) matching the frequency chirp induced in the first stage. The third, mixing stage, was built using a 240-m long dispersion-flattened HNLF possessing small normal dispersion. This section (HNLF2) was engineered with dispersive variation below 1 ps/nm/km over the entire comb bandwidth of 200 nm. The dispersion of the HNLF2 was precisely controlled by applying spatially constant tension to be well within the normal dispersion region, which effectively suppressed modulation-instability amplified noise [16,17].

The phase-uncorrelated architecture, shown in Fig. 2(b), was characterized in order to allow rigorous comparison with the phase-correlated scheme of primary interest in this work. For the uncorrelated case, two free-running ECLs with 400-GHz spacing were used as master oscillators, while the reminder of the mixer configuration was kept unchanged with respect to the phase-correlated comb generator.

4. Measurement results and comparisons

Measured 200- and 400-GHz spaced parametric combs spanning over 200 nm are shown in Figs. 4 and 5 , respectively, with more than 35-dB OSNR and less than 10-dB peak-to-peak spectral flatness over 160 nm. Detailed comb optimization techniques used to achieve this response can be found in Refs. 16 and 17. Linewidths of selected comb lines were measured at both Stokes and anti-Stokes bands (as marked in Figs. 4 and 5) by using the standard self-heterodyne method [38]. In Fig. 6(a) , measured phase noise spectra of the uncorrelated-phase setup (Fig. 2(b)) are shown with normalized powers. Not surprisingly, significant linewidth broadening can be observed with increase in comb-tone order for uncorrelated pump case. As an example, for the 25th-order tone, the FWHM linewidth measured by self-heterodyne technique at both spectral sides are approximately 2 MHz, which corresponds to 1-MHz true linewidth, when assuming a Lorentzian line-shape [38]. This result clearly indicates severe phase noise degradation: the tone linewidth has grown more than two orders of magnitude from the original 7-kHz ECL linewidth. While severe, the measured penalty is in full accordance with that reported in Ref. 16.

 figure: Fig. 4

Fig. 4 Optical spectrum of the generated 200-GHz spaced parametric comb with marked seeding wavelengths and measured comb lines.

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 figure: Fig. 5

Fig. 5 Optical spectrum of the generated 400-GHz spaced parametric comb with marked seeding wavelengths and measured comb lines.

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 figure: Fig. 6

Fig. 6 Measured output comb-line spectra measured by self-heterodyne detection (in linear scale) of different setups: 400-GHz spaced comb with (a) un-correlated and (b) correlated pump phases, and (c) 200-GHz comb with correlated pumps. Insets of Figs. (b) and (c) show zoomed-in spectra. The linewidth scaling ratios (compared to the seeding laser) of different setups (400-GHz combs with and without injection locking, respectively) are shown in (d) as a function of the comb-line order.

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In contrast to rapid linewidth scaling in case when mixer is driven by phase-uncorrelated pumps, nearly constant linewidths were measured when phase-correlated pump seeds were produced via injection locking, as shown in 6(b) and (c). From the insets (200-kHz span) one can clearly see the preservation of line shapes at the 25th- (400-GHz pitch) and 50th-order (200-GHz pitch) comb lines. The measurement unambiguously confirms the effectiveness of phase-correlated pumps in a cavity-less parametric OFC. A more direct comparison is shown in Fig. 6(d), where measured linewidth scaling-ratios of the 400-GHz combs are plotted against the comb-line order. The comb seeded by phase-uncorrelated pumps exhibits quadratic linewidth increase, while phase-correlated pumps strictly preserve the original linewidth. Strictly speaking, both linear and quadratic scaling contributed to measured linewidth in the phase-uncorrelated scenario, representing different phase-noise contributions of the slave laser, i.e. white and 1/fnoise components, as explained in Section 2 [29,30].

5. Mechanisms for the residual linewidth broadening

In an ideal case, the phase-noise spectra of the comb tones seeded by phase-correlated pumps should be perfectly preserved, regardless of the line order. However, a more careful inspection of the measurements shown in Figs. 6(b) and 6(c) reveals a spectral tail around narrow central part of the tone. The spectral power density of the spectral tail grows with increasing tone order, implying a complex noise evolution within the mixer generator. Indeed, the observed line reshaping is attributed to mixture of correlated and uncorrelated noise contributions, ultimately leading to a modified line shape with a delta-function-like peak and a slower rolloff wings. Multiple mechanisms lead to the residual phase decorrelation [39]: (1) length mismatch between two pump paths; (2) phase noise from the radio-frequency (RF) source used to generate the 25-GHz carrier, inducing initial phase decorrelation before the injection locking; (3) insufficient injection-locking ratio, which narrows the locking bandwidth and thus leads to imperfect phase locking; (4) inherent phase noise from the master and slave lasers. Finally, optical amplification that follows the injection locking as well as Raman-phonon induced noise generation in the HNLFs [31] may also contribute to phase decorrelation. Accordingly, a partially-improved performance can be achieved by applying (1) precise path-length matching, (2) low phase-noise RF signal generator, (3) increased master laser output, and (4) narrower-linewidth master and slave lasers.

The impairment mechanisms were mapped against operating conditions in experimental architecture. Firstly, the pump paths were matched to within a few centimeters, while the injection power level was −10 dBm, allowing for a locking bandwidth of a few hundreds of MHz. This combination was deemed to have negligible influence on the linewidth broadening observed above. Secondly, the master laser indeed possessed narrow, kHz-scale linewidth, while the slave laser was characterized by a sub-MHz linewidth: a mismatch that, if closed by using narrower-linewidth slave oscillators, could lead to lower degree of phase decorrelation. Thirdly, and most importantly, the RF source used in the experiments had possessed phase noise characterized with spectral power density of −110 and −130-dBc/Hz at 100-kHz and 1-MHz frequency offsets, respectively. This level of noise was more than sufficient to provide a finite contribution to partial phase noise decorrelation observed in experiments.

In order to verify the hypothesis that phase noise from the RF source will degrade the linewidth preservation of an injection-locked parametric comb, the noise spectra of the comb lines corresponding to different pump separation (comb pitch) are compared in Fig. 7 . If one assumes that noise performance is dominated by RF source contribution, one should observe decrease in phase correlation between two tones created by phase modulators as their frequency difference grows [39]. As a consequence, the injection locked comb with wider pump separation should exhibit higher phase noise floor for the same tone order, otherwise the phase noise spectra will be identical without considering the RF phase noise. The measurement shown in Fig. 7, reveals that the described behavior was indeed replicated when comb pitch is increased. Consequently, the observed phase noise pedestal was, at least partially attributed to the inherent phase instability of the RF signal generator in the coherence replicating step, i.e. before the parametric comb generation stage.

 figure: Fig. 7

Fig. 7 Comparisons of measured comb-line spectra with 200- and 400-GHz frequency pitches at the same comb-line orders.

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6. Conclusion

We have, for the first time, to the best of our knowledge, demonstrated cavity-less parametric comb generation with high phase fidelity over bandwidth exceeding 200 nm. CW-seeded comb was generated with variable frequency pitches of 200- and 400-GHz and characterized with respect to efficiency, bandwidth and phase noise. Unlike previously results in which spectral tone linewidth scaled with the comb tone order, the linewidth of the new comb was preserved. This was achieved by seeding the comb with two phase-correlated CW pumps. Phase correlation between two pumps was obtained by injection-locking two distinct slave lasers with two phase-modulated sidebands of a single laser oscillator, guaranteeing high degree of coherence. In contrast to a quadratic-linewidth scaling characteristic of the commonly used, free-running pump generation, the new technique exhibits well-preserved FWHM linewidths over the entire 200 nm. The impairment mechanisms that can lead to residual phase noise were identified and described.

Acknowledgments

This work is based in part on research sponsored by the Office of Naval Research (ONR). The authors would like to acknowledge Sumitomo Electric Industries for providing the HNLFs used in this work.

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39. F. Kéfélian, R. Gabet, and P. Gallion, “Characteristics of the phase noise correlation of injection locked lasers for RF signal generation and transmission,” Opt. Quantum Electron. 38(4-6), 467–478 (2006). [CrossRef]  

References

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  1. J. L. Hall, “Optical frequency measurement: 40 years of technology revolutions,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1136–1144 (2000).
    [Crossref]
  2. Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
    [Crossref] [PubMed]
  3. S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75(1), 325–342 (2003).
    [Crossref]
  4. S. A. Diddams, “The evolving optical frequency comb,” J. Opt. Soc. Am. B 27(11), B51–B60 (2010).
    [Crossref]
  5. J. Mandon, G. Guelachvili, and N. Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009).
    [Crossref]
  6. W. H. Oskay, S. A. Diddams, E. A. Donley, T. M. Fortier, T. P. Heavner, L. Hollberg, W. M. Itano, S. R. Jefferts, M. J. Delaney, K. Kim, F. Levi, T. E. Parker, and J. C. Bergquist, “Single-atom optical clock with high accuracy,” Phys. Rev. Lett. 97(2), 020801 (2006).
    [Crossref] [PubMed]
  7. Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
    [Crossref]
  8. T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011).
    [Crossref]
  9. D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s−1 line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5(6), 364–371 (2011).
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    [Crossref] [PubMed]
  11. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
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  12. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
    [Crossref] [PubMed]
  13. H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1325–1331 (2000).
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  14. A. Cerqueira Sodre, J. M. Chavez Boggio, A. A. Rieznik, H. E. Hernandez-Figueroa, H. L. Fragnito, and J. C. Knight, “Highly efficient generation of broadband cascaded four-wave mixing products,” Opt. Express 16(4), 2816–2828 (2008).
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  16. B. P. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Wavelength multicasting via frequency comb generation in a bandwidth-enhanced fiber optical parametric mixer,” J. Lightwave Technol. 29(23), 3515–3522 (2011).
    [Crossref]
  17. E. Myslivets, B. P. P. Kuo, N. Alic, and S. Radic, “Generation of wideband frequency combs by continuous-wave seeding of multistage mixers with synthesized dispersion,” Opt. Express 20(3), 3331–3344 (2012).
    [Crossref] [PubMed]
  18. C.-S. Brès, A. O. J. Wiberg, B. P. P. Kuo, N. Alic, and S. Radic, “Wavelength multicasting of 320Gb/s channel in self-seeded parametric amplifier,” IEEE Photon. Technol. Lett. 21(14), 1002–1004 (2009).
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  19. S. Radic, “Parametric signal processing,” IEEE J. Sel. Top. Quantum Electron. 18(2), 670–680 (2012).
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  20. B. P. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Laser coherence enhancement by extra-cavity parametric mixing,” in Optical Fiber Communications Conference (OFC), paper PDP5A.3 (2012).
  21. J. Kakande, R. Slavík, F. Parmigiani, P. Petropoulos, and D. Richardson, “Overcoming electronic limits to optical phase measurements with an optical phase-only amplifier,” in Optical Fiber Communications Conference (OFC), paper PDP5C.9 (2012).
  22. I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
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  23. J. Zhou, R. Hui, and N. Caponio, “Spectral linewidth and frequency chirp of four-wave mixing components in optical fibers,” IEEE Photon. Technol. Lett. 6(3), 434–436 (1994).
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  24. S. K. Korotky, P. B. Hansen, L. Eskildsen, and J. J. Veselka, “Efficient phase modulation scheme for suppressing stimulated Brillouin scattering, in International Conference Integrated Optics and Optical Fiber Communications (IOOC), paper WD2–1 (1995).
  25. S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photon. Technol. Lett. 15(5), 673–675 (2003).
    [Crossref]
  26. E. Myslivets, C. Lundström, J. M. Aparicio, S. Moro, A. O. J. Wiberg, C.-S. Bres, N. Alic, P. A. Andrekson, and S. Radic, “Spatial equalization of zero dispersion wavelength profiles in nonlinear fibers,” IEEE Photon. Technol. Lett. 21(24), 1807–1809 (2009).
    [Crossref]
  27. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
    [Crossref]
  28. F. C. Cruz, “Optical frequency combs generated by four-wave mixing in optical fibers for astrophysical spectrometer calibration and metrology,” Opt. Express 16(17), 13267–13275 (2008).
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  29. K. Petermann, Diode Modulation and Noise (Kluwer Academic, 1988), Chap. 7.
  30. P. Horak and W. H. Loh, “On the delayed self-heterodyne interferometric technique for determining the linewidth of fiber lasers,” Opt. Express 14(9), 3923–3928 (2006).
    [Crossref] [PubMed]
  31. Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18(3), 2884–2893 (2010).
    [Crossref] [PubMed]
  32. C. Buczek, R. J. Freiberg, and M. L. Skolnick, “Laser injection locking,” Proc. IEEE 61(10), 1411–1431 (1973).
    [Crossref]
  33. R. Weerasuriya, S. Sygletos, S. K. Ibrahim, R. Phelan, J. O’Carroll, B. Kelly, J. O’Gorman, and A. D. Ellis, “Generation of frequency symmetric signals from a BPSK input for phase sensitive amplification,” in Optical Fiber Communications Conference (OFC), paper OWT6 (2010).
  34. R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
    [Crossref]
  35. S. L. I. Olsson, B. Corcoran, C. Lundström, E. Tipsuwannakul, S. Sygletos, A. D. Ellis, Z. Tong, M. Karlsson, and P. A. Andrekson, “Optical injection-locking-based pump recovery for phase-sensitively amplified links,” in Optical Fiber Communications Conference (OFC), paper OW3C (2012).
  36. R. Slavík, F. Parmigiani, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, P. Petropoulos, and D. J. Richardson, “Stable and efficient generation of high repetition rate (>160 GHz) subpicosecond optical pulses,” IEEE Photon. Technol. Lett. 23(9), 540–542 (2011).
    [Crossref]
  37. R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. 18(6), 976–983 (1982).
    [Crossref]
  38. T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
    [Crossref]
  39. F. Kéfélian, R. Gabet, and P. Gallion, “Characteristics of the phase noise correlation of injection locked lasers for RF signal generation and transmission,” Opt. Quantum Electron. 38(4-6), 467–478 (2006).
    [Crossref]

2012 (2)

2011 (5)

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref] [PubMed]

B. P. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Wavelength multicasting via frequency comb generation in a bandwidth-enhanced fiber optical parametric mixer,” J. Lightwave Technol. 29(23), 3515–3522 (2011).
[Crossref]

T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011).
[Crossref]

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s−1 line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5(6), 364–371 (2011).
[Crossref]

R. Slavík, F. Parmigiani, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, P. Petropoulos, and D. J. Richardson, “Stable and efficient generation of high repetition rate (>160 GHz) subpicosecond optical pulses,” IEEE Photon. Technol. Lett. 23(9), 540–542 (2011).
[Crossref]

2010 (3)

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18(3), 2884–2893 (2010).
[Crossref] [PubMed]

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
[Crossref]

S. A. Diddams, “The evolving optical frequency comb,” J. Opt. Soc. Am. B 27(11), B51–B60 (2010).
[Crossref]

2009 (4)

J. Mandon, G. Guelachvili, and N. Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009).
[Crossref]

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

C.-S. Brès, A. O. J. Wiberg, B. P. P. Kuo, N. Alic, and S. Radic, “Wavelength multicasting of 320Gb/s channel in self-seeded parametric amplifier,” IEEE Photon. Technol. Lett. 21(14), 1002–1004 (2009).
[Crossref]

E. Myslivets, C. Lundström, J. M. Aparicio, S. Moro, A. O. J. Wiberg, C.-S. Bres, N. Alic, P. A. Andrekson, and S. Radic, “Spatial equalization of zero dispersion wavelength profiles in nonlinear fibers,” IEEE Photon. Technol. Lett. 21(24), 1807–1809 (2009).
[Crossref]

2008 (3)

2007 (1)

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

2006 (3)

W. H. Oskay, S. A. Diddams, E. A. Donley, T. M. Fortier, T. P. Heavner, L. Hollberg, W. M. Itano, S. R. Jefferts, M. J. Delaney, K. Kim, F. Levi, T. E. Parker, and J. C. Bergquist, “Single-atom optical clock with high accuracy,” Phys. Rev. Lett. 97(2), 020801 (2006).
[Crossref] [PubMed]

P. Horak and W. H. Loh, “On the delayed self-heterodyne interferometric technique for determining the linewidth of fiber lasers,” Opt. Express 14(9), 3923–3928 (2006).
[Crossref] [PubMed]

F. Kéfélian, R. Gabet, and P. Gallion, “Characteristics of the phase noise correlation of injection locked lasers for RF signal generation and transmission,” Opt. Quantum Electron. 38(4-6), 467–478 (2006).
[Crossref]

2003 (2)

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photon. Technol. Lett. 15(5), 673–675 (2003).
[Crossref]

S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75(1), 325–342 (2003).
[Crossref]

2002 (2)

Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

2000 (3)

J. L. Hall, “Optical frequency measurement: 40 years of technology revolutions,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1136–1144 (2000).
[Crossref]

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[Crossref] [PubMed]

H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1325–1331 (2000).
[Crossref]

1994 (1)

J. Zhou, R. Hui, and N. Caponio, “Spectral linewidth and frequency chirp of four-wave mixing components in optical fibers,” IEEE Photon. Technol. Lett. 6(3), 434–436 (1994).
[Crossref]

1982 (1)

R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. 18(6), 976–983 (1982).
[Crossref]

1980 (1)

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
[Crossref]

1973 (1)

C. Buczek, R. J. Freiberg, and M. L. Skolnick, “Laser injection locking,” Proc. IEEE 61(10), 1411–1431 (1973).
[Crossref]

Alic, N.

E. Myslivets, B. P. P. Kuo, N. Alic, and S. Radic, “Generation of wideband frequency combs by continuous-wave seeding of multistage mixers with synthesized dispersion,” Opt. Express 20(3), 3331–3344 (2012).
[Crossref] [PubMed]

B. P. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Wavelength multicasting via frequency comb generation in a bandwidth-enhanced fiber optical parametric mixer,” J. Lightwave Technol. 29(23), 3515–3522 (2011).
[Crossref]

C.-S. Brès, A. O. J. Wiberg, B. P. P. Kuo, N. Alic, and S. Radic, “Wavelength multicasting of 320Gb/s channel in self-seeded parametric amplifier,” IEEE Photon. Technol. Lett. 21(14), 1002–1004 (2009).
[Crossref]

E. Myslivets, C. Lundström, J. M. Aparicio, S. Moro, A. O. J. Wiberg, C.-S. Bres, N. Alic, P. A. Andrekson, and S. Radic, “Spatial equalization of zero dispersion wavelength profiles in nonlinear fibers,” IEEE Photon. Technol. Lett. 21(24), 1807–1809 (2009).
[Crossref]

Andrekson, P. A.

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18(3), 2884–2893 (2010).
[Crossref] [PubMed]

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
[Crossref]

E. Myslivets, C. Lundström, J. M. Aparicio, S. Moro, A. O. J. Wiberg, C.-S. Bres, N. Alic, P. A. Andrekson, and S. Radic, “Spatial equalization of zero dispersion wavelength profiles in nonlinear fibers,” IEEE Photon. Technol. Lett. 21(24), 1807–1809 (2009).
[Crossref]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

Aparicio, J. M.

E. Myslivets, C. Lundström, J. M. Aparicio, S. Moro, A. O. J. Wiberg, C.-S. Bres, N. Alic, P. A. Andrekson, and S. Radic, “Spatial equalization of zero dispersion wavelength profiles in nonlinear fibers,” IEEE Photon. Technol. Lett. 21(24), 1807–1809 (2009).
[Crossref]

Araujo-Hauck, C.

T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321(5894), 1335–1337 (2008).
[Crossref] [PubMed]

Becker, J.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s−1 line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5(6), 364–371 (2011).
[Crossref]

Ben Ezra, S.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s−1 line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5(6), 364–371 (2011).
[Crossref]

Bergquist, J. C.

T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011).
[Crossref]

W. H. Oskay, S. A. Diddams, E. A. Donley, T. M. Fortier, T. P. Heavner, L. Hollberg, W. M. Itano, S. R. Jefferts, M. J. Delaney, K. Kim, F. Levi, T. E. Parker, and J. C. Bergquist, “Single-atom optical clock with high accuracy,” Phys. Rev. Lett. 97(2), 020801 (2006).
[Crossref] [PubMed]

Bogris, A.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
[Crossref]

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18(3), 2884–2893 (2010).
[Crossref] [PubMed]

Bonk, R.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s−1 line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5(6), 364–371 (2011).
[Crossref]

Brar, K.

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photon. Technol. Lett. 15(5), 673–675 (2003).
[Crossref]

Bres, C.-S.

E. Myslivets, C. Lundström, J. M. Aparicio, S. Moro, A. O. J. Wiberg, C.-S. Bres, N. Alic, P. A. Andrekson, and S. Radic, “Spatial equalization of zero dispersion wavelength profiles in nonlinear fibers,” IEEE Photon. Technol. Lett. 21(24), 1807–1809 (2009).
[Crossref]

Brès, C.-S.

C.-S. Brès, A. O. J. Wiberg, B. P. P. Kuo, N. Alic, and S. Radic, “Wavelength multicasting of 320Gb/s channel in self-seeded parametric amplifier,” IEEE Photon. Technol. Lett. 21(14), 1002–1004 (2009).
[Crossref]

Buczek, C.

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W. H. Oskay, S. A. Diddams, E. A. Donley, T. M. Fortier, T. P. Heavner, L. Hollberg, W. M. Itano, S. R. Jefferts, M. J. Delaney, K. Kim, F. Levi, T. E. Parker, and J. C. Bergquist, “Single-atom optical clock with high accuracy,” Phys. Rev. Lett. 97(2), 020801 (2006).
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R. Slavík, F. Parmigiani, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, P. Petropoulos, and D. J. Richardson, “Stable and efficient generation of high repetition rate (>160 GHz) subpicosecond optical pulses,” IEEE Photon. Technol. Lett. 23(9), 540–542 (2011).
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R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
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W. H. Oskay, S. A. Diddams, E. A. Donley, T. M. Fortier, T. P. Heavner, L. Hollberg, W. M. Itano, S. R. Jefferts, M. J. Delaney, K. Kim, F. Levi, T. E. Parker, and J. C. Bergquist, “Single-atom optical clock with high accuracy,” Phys. Rev. Lett. 97(2), 020801 (2006).
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Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
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S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photon. Technol. Lett. 15(5), 673–675 (2003).
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D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s−1 line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5(6), 364–371 (2011).
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S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photon. Technol. Lett. 15(5), 673–675 (2003).
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T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321(5894), 1335–1337 (2008).
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T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
[Crossref] [PubMed]

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T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011).
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Rieznik, A. A.

Roeger, M.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s−1 line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5(6), 364–371 (2011).
[Crossref]

Rosenband, T.

T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011).
[Crossref]

Schellinger, T.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s−1 line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5(6), 364–371 (2011).
[Crossref]

Schmidt, W.

T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321(5894), 1335–1337 (2008).
[Crossref] [PubMed]

Schmogrow, R.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s−1 line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5(6), 364–371 (2011).
[Crossref]

Sjödin, M.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
[Crossref]

Skolnick, M. L.

C. Buczek, R. J. Freiberg, and M. L. Skolnick, “Laser injection locking,” Proc. IEEE 61(10), 1411–1431 (1973).
[Crossref]

Slavík, R.

R. Slavík, F. Parmigiani, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, P. Petropoulos, and D. J. Richardson, “Stable and efficient generation of high repetition rate (>160 GHz) subpicosecond optical pulses,” IEEE Photon. Technol. Lett. 23(9), 540–542 (2011).
[Crossref]

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
[Crossref]

Steinmetz, T.

T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321(5894), 1335–1337 (2008).
[Crossref] [PubMed]

Stentz, A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[Crossref] [PubMed]

Swann, W. C.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Sygletos, S.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
[Crossref]

Syvridis, D.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
[Crossref]

Taylor, J.

T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011).
[Crossref]

Tong, Z.

Udem, T.

T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321(5894), 1335–1337 (2008).
[Crossref] [PubMed]

Udem, Th.

Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

Vallaitis, T.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s−1 line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5(6), 364–371 (2011).
[Crossref]

Weerasuriya, R.

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
[Crossref]

Weiner, A. M.

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

Weingarten, K.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s−1 line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5(6), 364–371 (2011).
[Crossref]

Westlund, M.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

Wiberg, A. O. J.

E. Myslivets, C. Lundström, J. M. Aparicio, S. Moro, A. O. J. Wiberg, C.-S. Bres, N. Alic, P. A. Andrekson, and S. Radic, “Spatial equalization of zero dispersion wavelength profiles in nonlinear fibers,” IEEE Photon. Technol. Lett. 21(24), 1807–1809 (2009).
[Crossref]

C.-S. Brès, A. O. J. Wiberg, B. P. P. Kuo, N. Alic, and S. Radic, “Wavelength multicasting of 320Gb/s channel in self-seeded parametric amplifier,” IEEE Photon. Technol. Lett. 21(14), 1002–1004 (2009).
[Crossref]

Wilken, T.

T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321(5894), 1335–1337 (2008).
[Crossref] [PubMed]

Windeler, R. S.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288(5466), 635–639 (2000).
[Crossref] [PubMed]

Winter, M.

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s−1 line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5(6), 364–371 (2011).
[Crossref]

Yamamoto, S.

H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1325–1331 (2000).
[Crossref]

Ye, J.

S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75(1), 325–342 (2003).
[Crossref]

Zhou, J.

J. Zhou, R. Hui, and N. Caponio, “Spectral linewidth and frequency chirp of four-wave mixing components in optical fibers,” IEEE Photon. Technol. Lett. 6(3), 434–436 (1994).
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T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630–631 (1980).
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R. Lang, “Injection locking properties of a semiconductor laser,” IEEE J. Quantum Electron. 18(6), 976–983 (1982).
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IEEE J. Sel. Top. Quantum Electron. (4)

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

J. L. Hall, “Optical frequency measurement: 40 years of technology revolutions,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1136–1144 (2000).
[Crossref]

H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1325–1331 (2000).
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S. Radic, “Parametric signal processing,” IEEE J. Sel. Top. Quantum Electron. 18(2), 670–680 (2012).
[Crossref]

IEEE Photon. Technol. Lett. (5)

C.-S. Brès, A. O. J. Wiberg, B. P. P. Kuo, N. Alic, and S. Radic, “Wavelength multicasting of 320Gb/s channel in self-seeded parametric amplifier,” IEEE Photon. Technol. Lett. 21(14), 1002–1004 (2009).
[Crossref]

R. Slavík, F. Parmigiani, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, P. Petropoulos, and D. J. Richardson, “Stable and efficient generation of high repetition rate (>160 GHz) subpicosecond optical pulses,” IEEE Photon. Technol. Lett. 23(9), 540–542 (2011).
[Crossref]

J. Zhou, R. Hui, and N. Caponio, “Spectral linewidth and frequency chirp of four-wave mixing components in optical fibers,” IEEE Photon. Technol. Lett. 6(3), 434–436 (1994).
[Crossref]

S. Radic, C. J. McKinstrie, R. M. Jopson, J. C. Centanni, A. R. Chraplyvy, C. G. Jorgensen, K. Brar, and C. Headley, “Selective suppression of idler spectral broadening in two-pump parametric architectures,” IEEE Photon. Technol. Lett. 15(5), 673–675 (2003).
[Crossref]

E. Myslivets, C. Lundström, J. M. Aparicio, S. Moro, A. O. J. Wiberg, C.-S. Bres, N. Alic, P. A. Andrekson, and S. Radic, “Spatial equalization of zero dispersion wavelength profiles in nonlinear fibers,” IEEE Photon. Technol. Lett. 21(24), 1807–1809 (2009).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Nat. Photonics (6)

J. Mandon, G. Guelachvili, and N. Picqué, “Fourier transform spectroscopy with a laser frequency comb,” Nat. Photonics 3(2), 99–102 (2009).
[Crossref]

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011).
[Crossref]

D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit s−1 line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics 5(6), 364–371 (2011).
[Crossref]

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

R. Slavík, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrøm, R. Phelan, J. O'Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4(10), 690–695 (2010).
[Crossref]

Nature (1)

Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Quantum Electron. (1)

F. Kéfélian, R. Gabet, and P. Gallion, “Characteristics of the phase noise correlation of injection locked lasers for RF signal generation and transmission,” Opt. Quantum Electron. 38(4-6), 467–478 (2006).
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W. H. Oskay, S. A. Diddams, E. A. Donley, T. M. Fortier, T. P. Heavner, L. Hollberg, W. M. Itano, S. R. Jefferts, M. J. Delaney, K. Kim, F. Levi, T. E. Parker, and J. C. Bergquist, “Single-atom optical clock with high accuracy,” Phys. Rev. Lett. 97(2), 020801 (2006).
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C. Buczek, R. J. Freiberg, and M. L. Skolnick, “Laser injection locking,” Proc. IEEE 61(10), 1411–1431 (1973).
[Crossref]

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S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75(1), 325–342 (2003).
[Crossref]

Science (3)

T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, L. Pasquini, A. Manescau, S. D’Odorico, M. T. Murphy, T. Kentischer, W. Schmidt, and T. Udem, “Laser frequency combs for astronomical observations,” Science 321(5894), 1335–1337 (2008).
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T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332(6029), 555–559 (2011).
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[Crossref] [PubMed]

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J. M. C. Boggio, S. Moro, N. Alic, M. Karlsson, J. Bland-Hawthorn, and S. Radic, “Nearly octave-spanning cascaded four-wave-mixing generation in low dispersion highly nonlinear fiber,” in European Conference on Optical Communications (ECOC), paper 9.1.2 (2009).

B. P. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Laser coherence enhancement by extra-cavity parametric mixing,” in Optical Fiber Communications Conference (OFC), paper PDP5A.3 (2012).

J. Kakande, R. Slavík, F. Parmigiani, P. Petropoulos, and D. Richardson, “Overcoming electronic limits to optical phase measurements with an optical phase-only amplifier,” in Optical Fiber Communications Conference (OFC), paper PDP5C.9 (2012).

R. Weerasuriya, S. Sygletos, S. K. Ibrahim, R. Phelan, J. O’Carroll, B. Kelly, J. O’Gorman, and A. D. Ellis, “Generation of frequency symmetric signals from a BPSK input for phase sensitive amplification,” in Optical Fiber Communications Conference (OFC), paper OWT6 (2010).

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S. L. I. Olsson, B. Corcoran, C. Lundström, E. Tipsuwannakul, S. Sygletos, A. D. Ellis, Z. Tong, M. Karlsson, and P. A. Andrekson, “Optical injection-locking-based pump recovery for phase-sensitively amplified links,” in Optical Fiber Communications Conference (OFC), paper OW3C (2012).

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Figures (7)

Fig. 1
Fig. 1 Illustration of linewidth scaling in a two-pump seeded parametric comb with (a) un-correlated and (b) correlated pump phases.
Fig. 2
Fig. 2 Experimental setup of parametric comb generation seeded by (a) two injection locked CW lasers with fully correlated phase relations and 200 GHz and 400 GHz frequency pitches, and (b) two free-running seed injection locked CW lasers with un-correlated phase relation and 400 GHz frequency pitch. ML: master laser, SL: slave laser, PM: phase modulator, PS: phase shifter, POBF: programmable optical bandpass filter, PC: polarization controller, BPF: bandpass filter, WDM: wavelength division multiplexer.
Fig. 3
Fig. 3 Comparison of characterized (a) linewidths and (b) RIN spectra of the master ECL, free-running DFB laser and injection locked DFB laser, respectively.
Fig. 4
Fig. 4 Optical spectrum of the generated 200-GHz spaced parametric comb with marked seeding wavelengths and measured comb lines.
Fig. 5
Fig. 5 Optical spectrum of the generated 400-GHz spaced parametric comb with marked seeding wavelengths and measured comb lines.
Fig. 6
Fig. 6 Measured output comb-line spectra measured by self-heterodyne detection (in linear scale) of different setups: 400-GHz spaced comb with (a) un-correlated and (b) correlated pump phases, and (c) 200-GHz comb with correlated pumps. Insets of Figs. (b) and (c) show zoomed-in spectra. The linewidth scaling ratios (compared to the seeding laser) of different setups (400-GHz combs with and without injection locking, respectively) are shown in (d) as a function of the comb-line order.
Fig. 7
Fig. 7 Comparisons of measured comb-line spectra with 200- and 400-GHz frequency pitches at the same comb-line orders.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

θ a1,s1 =2 θ P 1 , P 2 θ P 2 , P 1
θ aN,sN =(N+1) θ P 1 , P 2 N θ P 2 , P 1 ,
δ v aN,sN = (N+1) 2 δ v P 1 , P 2 + N 2 δ v P 2 , P 1 ,
δ v aN,sN =(N+1)δ v P 1 , P 2 +Nδ v P 2 , P 1 ,
θ P 1 = θ P 2 + θ 0 ,
θ aN,sN = θ P 1 , P 2 ±N θ 0 ,

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