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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 22 — Oct. 22, 2012
  • pp: 24880–24885
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Analysis of the feed-forward method for the referencing of a CW laser to a frequency comb

D. Gatti, T. Sala, A. Gambetta, N. Coluccelli, G. Nunzi Conti, G. Galzerano, P. Laporta, and M. Marangoni  »View Author Affiliations


Optics Express, Vol. 20, Issue 22, pp. 24880-24885 (2012)
http://dx.doi.org/10.1364/OE.20.024880


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Abstract

We report on a comprehensive theoretical and experimental analysis of the feed-forward method for external frequency stabilization of a continuous wave laser against a frequency comb. Application of the method to a distributed feedback diode laser at 1.55 μm allows line narrowing from 800 to 10 kHz, with frequency noise reduction by more than 2 decades up to a Fourier frequency of 100 kHz and a maximum control bandwidth of 0.8 MHz. The results are consistent with a relative phase fluctuation of 1.4 rad rms, as limited by uncompensated high-frequency noise of the slave laser.

© 2012 OSA

1. Introduction

Frequency-stabilized, narrow-linewidth lasers are fundamental tools in a wide range of applications such as optical frequency standards, high-resolution molecular spectroscopy and gas sensing, as well as optical communications. With the invention of optical frequency combs [1

1. S. A. Diddams, “The evolving optical frequency comb,” J. Opt. Soc. Am. B 27(11), B51–B62 (2010). [CrossRef]

], the world of spectroscopy has benefited from an unprecedented tool, which allows CW lasers to be referenced to a highly repeatable and absolute frequency axis. Repeatability is a key feature whenever highly accurate absorption profiles have to be measured, enabling the acquisition of traceable spectroscopic data and the study of the physics underlying line-broadening mechanisms [2

2. V. Ahtee, M. Merimaa, and K. Nyholm, “Precision spectroscopy of acetylene transitions using an optical frequency synthesizer,” Opt. Lett. 34(17), 2619–2621 (2009). [CrossRef] [PubMed]

4

4. C. P. McRaven, M. J. Cich, G. V. Lopez, T. J. Sears, D. Hurtmans, and A. W. Mantz, “Frequency comb-referenced measurements of self- and nitrogen-broadening in the ν1 + ν3 band of acetylene,” J. Mol. Spectrosc. 266(1), 43–51 (2011). [CrossRef]

]. On the other hand, absolute frequency calibration enables the comparison of spectroscopic data acquired in different laboratories and at different times [5

5. C. G. Parthey, A. Matveev, J. Alnis, B. Bernhardt, A. Beyer, R. Holzwarth, A. Maistrou, R. Pohl, K. Predehl, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, and T. W. Hänsch, “Improved Measurement of the Hydrogen 1S-2S Transition Frequency,” Phys. Rev. Lett. 107(20), 203001 (2011). [CrossRef] [PubMed]

], as well as comparison with theoretical predictions or existing databases [6

6. F. L. Hong, A. Onae, J. Jiang, R. Guo, H. Inaba, K. Minoshima, T. R. Schibli, H. Matsumoto, and K. Nakagawa, “Absolute frequency measurement of an acetylene-stabilized laser at 1542 nm,” Opt. Lett. 28(23), 2324–2326 (2003). [CrossRef] [PubMed]

].

A common limitation to several experiments comes from the frequency noise of the probing laser that is usually higher than that of combs. Transferring the coherence properties from the comb to the probe laser allows to circumvent such a limitation and to obtain absolute frequency calibration. The traditional way to obtain coherence transfer from a master laser to a slave laser relies on the realisation of a robust and fast servo loop [7

7. J. D. Jost, J. L. Hall, and J. Ye, “Continuously tunable, precise, single frequency optical signal generator,” Opt. Express 10(12), 515–520 (2002). [PubMed]

] providing feedback to the laser through proper frequency actuators, e.g. piezo-transducers (PZT) and/or current modulation ports. However, this task requires a careful design of the servo [8

8. L. Matos, O. D. Mücke, J. Chen, and F. X. Kärtner, “Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers,” Opt. Express 14(6), 2497–2511 (2006). [CrossRef] [PubMed]

]. Moreover, wide-bandwidth feedback control becomes impracticable in the absence of fast frequency modulation ports such as bias-T or FET ports. In such cases, the use of an acousto-optic modulator in a feedback loop can solve the problem [9

9. J. L. Hall and T. W. Hänsch, “External dye-laser frequency stabilizer,” Opt. Lett. 9(11), 502–504 (1984). [CrossRef] [PubMed]

]. Very recently [10

10. S. Koke, C. Grebing, H. Frei, A. Anderson, A. Assion, and G. Steinmeyer, “Direct frequency comb synthesis with arbitrary offset and shot-noise-limited phase noise,” Nat. Photonics 4(7), 462–465 (2010). [CrossRef]

], an alternative approach based on an acousto-optic-frequency shifter (AOFS) in a feed-forward configuration has been shown to be successful both to obtain carrier-envelope phase (CEP) stabilization with extremely low residual phase jitter [11

11. F. Lücking, A. Assion, A. Apolonski, F. Krausz, and G. Steinmeyer, “Long-term carrier-envelope-phase-stable few-cycle pulses by use of the feed-forward method,” Opt. Lett. 37(11), 2076–2078 (2012). [CrossRef] [PubMed]

] and to achieve comb referencing and line-narrowing of extended cavity diode lasers [12

12. T. Sala, D. Gatti, A. Gambetta, N. Coluccelli, G. Galzerano, P. Laporta, and M. Marangoni, “Wide-bandwidth phase lock between a CW laser and a frequency comb based on a feed-forward configuration,” Opt. Lett. 37(13), 2592–2594 (2012). [CrossRef] [PubMed]

]. These papers, although showing significant reduction of phase and frequency noise, do not provide details on the maximum control bandwidth and on the amount of coherence transfer that can be achieved in the presence of high-frequency laser noise.

In this paper, a comprehensive theoretical and experimental analysis of the frequency response of the feed-forward method is provided, which is a crucial point to assess the control bandwidth and more generally the practicability and performance of the method. The experimental validation has been carried out by referencing and narrowing a distributed feedback (DFB) laser at 1.55 μm against an Er-based frequency comb. The DFB laser exhibits a frequency noise spectrum dominated by a white noise contribution and a linewidth as large as 800 kHz on a 1 ms time-scale. As such it represents a more demanding case with respect to that considered by Koke et al. for the stabilization of the CEP of a Ti:sapphire comb and by Sala et al. for the stabilization of an extended cavity diode laser. Theoretical and experimental results attest the control bandwidth to be limited to 1/6 of the inverse time response of the AOFS, translating into 800 kHz in the specific case of our AOFS, whose minimum response time is 210 ns. Such regime is shown to be sufficient for a consistent line-narrowing of the laser against the frequency comb, but not to push the residual rms phase jitter below 1.4 rad due to a cosine transfer function that does not allow for an efficient noise reduction above 100 kHz. If considering the inherent robustness and simplicity of the method the results are of particular interest for comb-assisted spectroscopy and in general to obtain a straightforward referencing of a slave to a master laser.

2. Feed-forward operation principle and control bandwidth

The analysis of the feed-forward method can be carried out, without any loss of generality, considering a frequency comb as a master laser. The operation principle requires the beating signal fbeat=|νnνcw| between the cw slave laser frequency νcw and the nearest comb mode νn to be used as the driving signal for an external AOFS, in such a way that fAOFS=fbeat. By properly choosing the sign of the diffraction order, or alternatively the sign of fbeat, the frequency of the 1st order diffracted beam results to collapse to νn according to the relation ν1st=νcw+νnνcw=νn. In other words the AOFS corrects any frequency deviation of the cw laser with respect to the comb and the result is the coincidence of their frequencies. This is a clear advantage with respect to an electronic feedback, where an offset frequency between master and slave is typical. An offset frequency locking is anyway possible by mixing fbeat with a radiofrequency fLO provided by a local oscillator. In this way the AOFS driving frequency becomes fAOFS=fbeat+fLO, giving ν1st=νn+fLO. It is worth emphasizing that in order to preserve fAOFS from drifting outside the AOFS working range, fbeat needs to be stabilized with a conventional slow feedback loop around a predetermined valueΔf0, with Δf0 and fLO obeying the relation Δf0+fLO=f¯AOFS, where f¯AOFS is the central operation frequency of the AOFS.

Since the frequency correction does not involve any actuator of the laser source, the only limitation to the control bandwidth arises from the time delay Δ between the acquisition of fbeat(t) and its effect on the beam traveling through the AOFS. This time is dictated both by the time the acoustic wave takes to reach and cross the optical beam inside the AOFS and the time lag introduced by the electronics. When taking into account such delay, ν1stbecomes:
ν1st(t)=νcw(t)+fbeat(tΔ)=[νcw(t)ν1st(tΔ)]+νn(tΔ)
(1)
The corresponding first order autocorrelation function is then given by:
R1st(τ)=ν1st(t)ν1st(t+τ)=νcw(t)νcw(t+τ)+νcw(tΔ)νcw(tΔ+τ)++νn(tΔ)νn(tΔ+τ)νcw(t)νcw(tΔ+τ)νcw(tΔ)νcw(t+τ)==2Rcw(τ)+Rn(τ)Rcw(τΔ)Rcw(τ+Δ)
(2)
where cross terms between comb and cw-laser have been neglected since uncorrelated. The spectral density S1stν(f)of the frequency fluctuations of the first order diffracted beam is then calculated from the Fourier transform of the autocorrelation function:
S1stν(f)={R1st(τ)}=2Scwν(f)+Snν(f)Scwν(f)ei2πfΔScwν(f)ei2πfΔ
(3)
which simplifies into:
S1stν(f)=Snν(f)+2Scwν(f)(1cos2πfΔ)
(4)
By expanding the cosine function in Mc Laurin series, Eq. (4) shows that the diffracted beam inherits the spectral properties of the comb as long as 2π2f2Δ21. As an estimator of the control bandwidth B we will use the Fourier frequency at which S1stν(f)matches Scwν(f):
B=16Δ
(5)
As it will be shown in the following paragraph, Eq. (5) very well predicts the experimental behaviour. It can be thus directly used for a quick estimate of the control bandwidth whenever the time delay Δ introduced by the AOFS is known.

3. Experimental setup

The experimental setup is shown in Fig. 1
Fig. 1 Experimental setup. AOFS, acousto-optical frequency shifter; BS, beam splitter; SM, spherical mirror; DM, dichroic mirror; G, grating; PD, photodetector; HVA, RF power amplifier; PBS, polarizer beam splitter; FD, frequency divider; ESA, electrical spectrum analyser.
. The slave laser is a single-mode DFB cw laser emitting up to 80 mW of power at 1.547 μm and is referred to a master laser constituted by a near-infrared frequency comb generated by a 100-MHz Er:fiber femtosecond oscillator. After spectral filtering of the comb spectrum, the two beams are superimposed to each other on an amplified PIN detector with a 125-MHz bandwidth. The resulting beat-note signal is roughly stabilized at 20 MHz by use of a slow feedback loop with a 7 kHz bandwidth acting on the DFB laser current. After mixing with an RF local oscillator at 180 MHz, is shifted to 160-MHz and then divided by two to match the AOFS working range centred at 80 MHz. The division by two is motivated by the adoption of a cat-eye configuration [13

13. E. A. Donley, T. P. Heavner, F. Levi, M. O. Tataw, and S. R. Jefferts, “Double-pass acousto-optic modulator system,” Rev. Sci. Instrum. 76(6), 063112 (2005). [CrossRef]

] that helps in reducing angular flickering of the diffracted beam induced by residual frequency jitter of . Such scheme employs a double pass configuration where the beam is reflected backward into the AOFS by a spherical mirror, with a combination of quarter wave-plate and polarizing beam splitter to allow extraction of the beam with negligible insertion losses.

The experimental setup is equipped with a self-heterodyne interferometer for the direct measurement of the laser lineshapes. It includes an acousto-optic modulator separating the two beams replica by an amount of 40 MHz and a 130-km-long fiber delay stage. For the measurement of the frequency noise of the DFB laser in different regimes, with and without feed-forward, a Fabry–Perot (FP) frequency discriminator is used. With a finesse of ~350 and a free-spectral-range of 5.9 GHz, it provides a frequency-to-amplitude conversion factor of about 23 nV/Hz.

4. Results and discussion

Preliminarily to lineshape and frequency noise measurements, the time delay introduced by the AOFS was evaluated, this parameter being critical for the effective control bandwidth. Figure 2(a)
Fig. 2 (a) Time response of the AOFS for different displacements of the beam from the piezo-actuator. (b) Frequency noise spectral density of the DFB slave laser in free running (FR) and in feed-forward (FF) regime for time delays corresponding to those of Fig. 2(a).
shows the time response of the AOFS as obtained by modulating the diffraction efficiency with a square-wave signal. The curves refer to different displacements of the beam from the PZT. In the best conditions, when the beam is as close as possible to the PZT, the time delay attains 210 ns. This value is consistent with the time needed for the acoustic wave to cross an optical beam with a ~100-μm diameter, which amounts to 40 ns, with an experimentally determined electronic time lag of 110 ns, and with a residual delay due to the displacement of the beam from the actuator. For such delay Eq. (5) predicts a control bandwidth as high as 795 kHz.

Figure 3(a)
Fig. 3 Self-heterodyne spectra in different regimes with logarithmic (a) and linear (b) scale. The inset refers to feed-forward at the fastest control bandwidth.
reports the lineshapes provided by the self-heterodyne interferometer at the different control bandwidths, as acquired with a resolution bandwidth (RBW) of 10 kHz. The linewidth shrinks significantly upon increasing the bandwidth, from 830 down to 8 kHz in the conditions of faster locking, as better evidenced by the linear traces reported in Fig. 3(b). The self-consistency of the acquired data was verified by comparing the experimental linewidths with those calculated by integration of the noise spectra of Fig. 2(b) (see formula on Ref. 14

14. G. Di Domenico, S. Schilt, and P. Thomann, “Simple approach to the relation between laser frequency noise and laser line shape,” Appl. Opt. 49(25), 4801–4807 (2010). [CrossRef] [PubMed]

). A minimum integration frequency of 1 kHz, accounting for a 1 ms observation time, was used to match the self-heterodyne interferometer time scale. The results reported in Table 1

Table 1. Calculated and measured full-width-at-half-maximum (FWHM) linewidths in different regimes.

table-icon
View This Table
exhibit a satisfactory agreement: the linewidths retrieved from the FP discriminator result slightly in excess because of the technical noise introduced by the FP cavity.

As a last test we directly observed the beating of the first order diffracted beam with the comb, in order to get a deeper insight into the amount of phase coherence transfer. The beat note spectrum is reported in Fig. 4
Fig. 4 Beating signal between comb and DFB laser in the feed-forward regime at decreasing RBW from 300 to 10 Hz. Inset: normalized beating signal at 10 Hz RBW.
over a span of 8 MHz for decreasing RBW from 300 kHz to 10 Hz. The spectrum clearly presents a central peak with a RBW-limited linewidth down to the lower spectral resolution of the analyser. The fractional power within such peak remains as high as 50% for RBW > 10 kHz, but drops down to 14% at RBW = 10 Hz. This behaviour reflects the presence of uncompensated frequency noise. This mostly occurs at high Fourier frequencies, where the feed-forward method fails to feed power into the coherent carrier. According to Ref. 15

15. J. L. Hall and M. Zhu, “An Introduction to Phase Stable Optical Sources,” in Laser Manipulation of Atoms and Ions, E. Arimondo, W. D. Phillips, and F. Strumia, eds. (North Holland, 1992), 671.

the residual rms phase fluctuation amounts to 1.4 rad. Bringing the coherence transfer to a sub-radiant level would be possible with slave lasers having a lower content of high-frequency noise. Two representative examples in such sense are quantum cascade lasers, which are characterized by flicker noise up to frequencies > 1 MHz [16

16. S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19(19), 17996–18003 (2011). [CrossRef] [PubMed]

,17

17. A. A. Mills, D. Gatti, J. Jiang, C. Mohr, W. Mefford, L. Gianfrani, M. Fermann, I. Hartl, and M. Marangoni, “Coherent phase lock of a 9 μm quantum cascade laser to a 2 μm thulium optical frequency comb,” Opt. Lett. 37(19), 4083–4085 (2012). [CrossRef] [PubMed]

] and the frequency combs themselves, in those application where a tight phase-lock with kHz or sub-kHz single line lasers is to be achieved.

5. Outlook and conclusions

The paper demonstrates the feed-forward method as a valuable and straightforward tool to obtain referencing of a slave to a master laser with a control bandwidth that can approach the MHz level for a sub-200 ns time response of the AOFS, thus useful to follow even fast frequency sweeps of the slave laser. As compared to a more traditional scheme using an AOFS in a feed-back configuration the main advantage is that no tuning of the loop parameters is needed. The results obtained widen the potential of the method, as originally proposed for carrier-envelope phase stabilization only. It can indeed be applied both to obtain tight frequency locking of a slave laser to a frequency comb and in a reverse mode to transfer the coherence of an optical standard to the frequency comb itself wherever the comb source lacks an intra-cavity electro-optical modulator.

Acknowledgments

The authors acknowledge support from a project of Regione Lombardia (CUP No. D41J0000300007).

References and links

1.

S. A. Diddams, “The evolving optical frequency comb,” J. Opt. Soc. Am. B 27(11), B51–B62 (2010). [CrossRef]

2.

V. Ahtee, M. Merimaa, and K. Nyholm, “Precision spectroscopy of acetylene transitions using an optical frequency synthesizer,” Opt. Lett. 34(17), 2619–2621 (2009). [CrossRef] [PubMed]

3.

A. Gambetta, D. Gatti, A. Castrillo, G. Galzerano, P. Laporta, L. Gianfrani, and M. Marangoni, “Mid-infrared quantitative spectroscopy by comb-referencing of a quantum-cascade-laser: Application to the CO2 spectrum at 4.3 μm,” Appl. Phys. Lett. 99(25), 251107 (2011). [CrossRef]

4.

C. P. McRaven, M. J. Cich, G. V. Lopez, T. J. Sears, D. Hurtmans, and A. W. Mantz, “Frequency comb-referenced measurements of self- and nitrogen-broadening in the ν1 + ν3 band of acetylene,” J. Mol. Spectrosc. 266(1), 43–51 (2011). [CrossRef]

5.

C. G. Parthey, A. Matveev, J. Alnis, B. Bernhardt, A. Beyer, R. Holzwarth, A. Maistrou, R. Pohl, K. Predehl, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, and T. W. Hänsch, “Improved Measurement of the Hydrogen 1S-2S Transition Frequency,” Phys. Rev. Lett. 107(20), 203001 (2011). [CrossRef] [PubMed]

6.

F. L. Hong, A. Onae, J. Jiang, R. Guo, H. Inaba, K. Minoshima, T. R. Schibli, H. Matsumoto, and K. Nakagawa, “Absolute frequency measurement of an acetylene-stabilized laser at 1542 nm,” Opt. Lett. 28(23), 2324–2326 (2003). [CrossRef] [PubMed]

7.

J. D. Jost, J. L. Hall, and J. Ye, “Continuously tunable, precise, single frequency optical signal generator,” Opt. Express 10(12), 515–520 (2002). [PubMed]

8.

L. Matos, O. D. Mücke, J. Chen, and F. X. Kärtner, “Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers,” Opt. Express 14(6), 2497–2511 (2006). [CrossRef] [PubMed]

9.

J. L. Hall and T. W. Hänsch, “External dye-laser frequency stabilizer,” Opt. Lett. 9(11), 502–504 (1984). [CrossRef] [PubMed]

10.

S. Koke, C. Grebing, H. Frei, A. Anderson, A. Assion, and G. Steinmeyer, “Direct frequency comb synthesis with arbitrary offset and shot-noise-limited phase noise,” Nat. Photonics 4(7), 462–465 (2010). [CrossRef]

11.

F. Lücking, A. Assion, A. Apolonski, F. Krausz, and G. Steinmeyer, “Long-term carrier-envelope-phase-stable few-cycle pulses by use of the feed-forward method,” Opt. Lett. 37(11), 2076–2078 (2012). [CrossRef] [PubMed]

12.

T. Sala, D. Gatti, A. Gambetta, N. Coluccelli, G. Galzerano, P. Laporta, and M. Marangoni, “Wide-bandwidth phase lock between a CW laser and a frequency comb based on a feed-forward configuration,” Opt. Lett. 37(13), 2592–2594 (2012). [CrossRef] [PubMed]

13.

E. A. Donley, T. P. Heavner, F. Levi, M. O. Tataw, and S. R. Jefferts, “Double-pass acousto-optic modulator system,” Rev. Sci. Instrum. 76(6), 063112 (2005). [CrossRef]

14.

G. Di Domenico, S. Schilt, and P. Thomann, “Simple approach to the relation between laser frequency noise and laser line shape,” Appl. Opt. 49(25), 4801–4807 (2010). [CrossRef] [PubMed]

15.

J. L. Hall and M. Zhu, “An Introduction to Phase Stable Optical Sources,” in Laser Manipulation of Atoms and Ions, E. Arimondo, W. D. Phillips, and F. Strumia, eds. (North Holland, 1992), 671.

16.

S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express 19(19), 17996–18003 (2011). [CrossRef] [PubMed]

17.

A. A. Mills, D. Gatti, J. Jiang, C. Mohr, W. Mefford, L. Gianfrani, M. Fermann, I. Hartl, and M. Marangoni, “Coherent phase lock of a 9 μm quantum cascade laser to a 2 μm thulium optical frequency comb,” Opt. Lett. 37(19), 4083–4085 (2012). [CrossRef] [PubMed]

OCIS Codes
(120.3930) Instrumentation, measurement, and metrology : Metrological instrumentation
(230.1040) Optical devices : Acousto-optical devices
(300.3700) Spectroscopy : Linewidth

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: August 14, 2012
Revised Manuscript: October 8, 2012
Manuscript Accepted: October 9, 2012
Published: October 16, 2012

Citation
D. Gatti, T. Sala, A. Gambetta, N. Coluccelli, G. Nunzi Conti, G. Galzerano, P. Laporta, and M. Marangoni, "Analysis of the feed-forward method for the referencing of a CW laser to a frequency comb," Opt. Express 20, 24880-24885 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24880


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References

  1. S. A. Diddams, “The evolving optical frequency comb,” J. Opt. Soc. Am. B27(11), B51–B62 (2010). [CrossRef]
  2. V. Ahtee, M. Merimaa, and K. Nyholm, “Precision spectroscopy of acetylene transitions using an optical frequency synthesizer,” Opt. Lett.34(17), 2619–2621 (2009). [CrossRef] [PubMed]
  3. A. Gambetta, D. Gatti, A. Castrillo, G. Galzerano, P. Laporta, L. Gianfrani, and M. Marangoni, “Mid-infrared quantitative spectroscopy by comb-referencing of a quantum-cascade-laser: Application to the CO2 spectrum at 4.3 μm,” Appl. Phys. Lett.99(25), 251107 (2011). [CrossRef]
  4. C. P. McRaven, M. J. Cich, G. V. Lopez, T. J. Sears, D. Hurtmans, and A. W. Mantz, “Frequency comb-referenced measurements of self- and nitrogen-broadening in the ν1 + ν3 band of acetylene,” J. Mol. Spectrosc.266(1), 43–51 (2011). [CrossRef]
  5. C. G. Parthey, A. Matveev, J. Alnis, B. Bernhardt, A. Beyer, R. Holzwarth, A. Maistrou, R. Pohl, K. Predehl, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Salomon, P. Laurent, and T. W. Hänsch, “Improved Measurement of the Hydrogen 1S-2S Transition Frequency,” Phys. Rev. Lett.107(20), 203001 (2011). [CrossRef] [PubMed]
  6. F. L. Hong, A. Onae, J. Jiang, R. Guo, H. Inaba, K. Minoshima, T. R. Schibli, H. Matsumoto, and K. Nakagawa, “Absolute frequency measurement of an acetylene-stabilized laser at 1542 nm,” Opt. Lett.28(23), 2324–2326 (2003). [CrossRef] [PubMed]
  7. J. D. Jost, J. L. Hall, and J. Ye, “Continuously tunable, precise, single frequency optical signal generator,” Opt. Express10(12), 515–520 (2002). [PubMed]
  8. L. Matos, O. D. Mücke, J. Chen, and F. X. Kärtner, “Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers,” Opt. Express14(6), 2497–2511 (2006). [CrossRef] [PubMed]
  9. J. L. Hall and T. W. Hänsch, “External dye-laser frequency stabilizer,” Opt. Lett.9(11), 502–504 (1984). [CrossRef] [PubMed]
  10. S. Koke, C. Grebing, H. Frei, A. Anderson, A. Assion, and G. Steinmeyer, “Direct frequency comb synthesis with arbitrary offset and shot-noise-limited phase noise,” Nat. Photonics4(7), 462–465 (2010). [CrossRef]
  11. F. Lücking, A. Assion, A. Apolonski, F. Krausz, and G. Steinmeyer, “Long-term carrier-envelope-phase-stable few-cycle pulses by use of the feed-forward method,” Opt. Lett.37(11), 2076–2078 (2012). [CrossRef] [PubMed]
  12. T. Sala, D. Gatti, A. Gambetta, N. Coluccelli, G. Galzerano, P. Laporta, and M. Marangoni, “Wide-bandwidth phase lock between a CW laser and a frequency comb based on a feed-forward configuration,” Opt. Lett.37(13), 2592–2594 (2012). [CrossRef] [PubMed]
  13. E. A. Donley, T. P. Heavner, F. Levi, M. O. Tataw, and S. R. Jefferts, “Double-pass acousto-optic modulator system,” Rev. Sci. Instrum.76(6), 063112 (2005). [CrossRef]
  14. G. Di Domenico, S. Schilt, and P. Thomann, “Simple approach to the relation between laser frequency noise and laser line shape,” Appl. Opt.49(25), 4801–4807 (2010). [CrossRef] [PubMed]
  15. J. L. Hall and M. Zhu, “An Introduction to Phase Stable Optical Sources,” in Laser Manipulation of Atoms and Ions, E. Arimondo, W. D. Phillips, and F. Strumia, eds. (North Holland, 1992), 671.
  16. S. Bartalini, S. Borri, I. Galli, G. Giusfredi, D. Mazzotti, T. Edamura, N. Akikusa, M. Yamanishi, and P. De Natale, “Measuring frequency noise and intrinsic linewidth of a room-temperature DFB quantum cascade laser,” Opt. Express19(19), 17996–18003 (2011). [CrossRef] [PubMed]
  17. A. A. Mills, D. Gatti, J. Jiang, C. Mohr, W. Mefford, L. Gianfrani, M. Fermann, I. Hartl, and M. Marangoni, “Coherent phase lock of a 9 μm quantum cascade laser to a 2 μm thulium optical frequency comb,” Opt. Lett.37(19), 4083–4085 (2012). [CrossRef] [PubMed]

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