## Nonlinear amplification of side-modes in frequency combs |

Optics Express, Vol. 21, Issue 10, pp. 11670-11687 (2013)

http://dx.doi.org/10.1364/OE.21.011670

Acrobat PDF (1404 KB)

### Abstract

We investigate how suppressed modes in frequency combs are modified upon frequency doubling and self-phase modulation. We find, both experimentally and by using a simplified model, that these side-modes are amplified relative to the principal comb modes. Whereas frequency doubling increases their relative strength by 6 dB, the growth due to self-phase modulation can be much stronger and generally increases with nonlinear propagation length. Upper limits for this effect are derived in this work. This behavior has implications for high-precision calibration of spectrographs with frequency combs used for example in astronomy. For this application, Fabry-Pérot filter cavities are used to increase the mode spacing to exceed the resolution of the spectrograph. Frequency conversion and/or spectral broadening after non-perfect filtering reamplify the suppressed modes, which can lead to calibration errors.

© 2013 OSA

## 1. Introduction

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

2. M. Mayor and D. Queloz, “A Jupiter-mass companion to a solar-type star,” Nature **378**, 355–359 (1995) [CrossRef] .

3. J. Schneider, “The extrasolar planets encyclopaedia,” http://exoplanet.eu/catalog.php.

4. J. Liske, A. Grazian, E. Vanzella, M. Dessauges, M. Viel, L. Pasquini, M. Haehnelt, S. Cristiani, F. Pepe, G. Avila, P. Bonifacio, F. Bouchy, H. Dekker, B. Delabre, S. D’Odorico, V. D’Odorico, S. Levshakov, C. Lovis, M. Mayor, P. Molaro, L. Moscardini, M. T. Murphy, D. Queloz, P. Shaver, S. Udry, T. Wiklind, and S. Zucker, “Cosmic dynamics in the era of extremely large telescopes,” Mon. Not. R. Astron. Soc. **386**, 1192–1218 (2008) [CrossRef] .

5. M. T. Murphy, Th. Udem, R. Holzwarth, A. Sizmann, L. Pasquini, C. Araujo-Hauck, H. Dekker, S. D’Odorico, M. Fischer, T. W. Hänsch, and A. Manescau, “High-precision wavelength calibration of astronomical spectrographs with laser frequency combs,” Mon. Not. R. Astron. Soc. **380**, 839–847 (2007) [CrossRef] .

6. A. Bartels, D. Heinecke, and S. A. Diddams, “10-GHz self-referenced optical frequency comb,” Science **326**, 681 (2009) [CrossRef] [PubMed] .

7. J. J. McFerran, L. Nenadovic, W. C. Swann, J. B. Schlager, and N. R. Newbury, “A passively mode-locked fiber laser at 1.54 *μ*m with a fundamental repetition frequency reaching 2 GHz,” Opt. Express **15**, 13155–13166 (2007) [CrossRef] [PubMed] .

8. H.-W. Chen, G. Chang, S. Xu, Z. Yang, and F. X. Kärtner, “3 GHz, fundamentally mode-locked, femtosecond Yb-fiber laser,” Opt. Lett. **37**, 3522–3524 (2012) [CrossRef] [PubMed] .

9. 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 Th. Udem, “Laser frequency combs for astronomical observations,” Science **321**, 1335–1337 (2008) [CrossRef] [PubMed] .

12. T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, and Th. Udem, “Fabry-Perot filter cavities for wide-spaced frequency combs with large spectral bandwidth,” Appl. Phys. B **96**, 251–256 (2009) [CrossRef] .

*m*of the mode spacing of the frequency comb and stabilized to it. This suppresses all but every

*m*mode of the initial frequency comb, multiplying the repetition rate accordingly (see Fig. 1). The filter cavity can be stabilized for example by locking it to a continuous wave laser, that itself is locked to one of the modes of the initial frequency comb. For a perfectly mode-matched, dispersion compensated cavity with finesse

^{th}*F*, the suppression factor of the closest mode is given approximately by (2

*F/m*)

^{2}. Limited side-mode suppression can lead to calibration errors, if the side-modes are not spectrally resolved [11

11. D. A. Braje, M. S. Kirchner, S. Osterman, T. Fortier, and S. A. Diddams, “Astronomical spectrograph calibration with broad-spectrum frequency combs,” Eur. Phys. J. D **48**, 57–66 (2008) [CrossRef] .

13. G. Chang, C.-H. Li, D. F. Phillips, R. L. Walsworth, and F. X. Kaertner, “Toward a broadband astro-comb: effects of nonlinear spectral broadening in optical fibers,” Opt. Express **18**, 12736–12747 (2010) [CrossRef] [PubMed] .

14. C.-H. Li, A. G. Glenday, A. J. Benedick, G. Chang, L.-J. Chen, C. Cramer, P. Fendel, G. Furesz, F. X. Kärtner, S. Korzennik, D. F. Phillips, D. Sasselov, A. Szentgyorgyi, and R. L. Walsworth, “In-situ determination of astro-comb calibrator lines to better than 10 cm s^{−1},” Opt. Express **18**, 13239–13249 (2010) [CrossRef] [PubMed] .

11. D. A. Braje, M. S. Kirchner, S. Osterman, T. Fortier, and S. A. Diddams, “Astronomical spectrograph calibration with broad-spectrum frequency combs,” Eur. Phys. J. D **48**, 57–66 (2008) [CrossRef] .

12. T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, and Th. Udem, “Fabry-Perot filter cavities for wide-spaced frequency combs with large spectral bandwidth,” Appl. Phys. B **96**, 251–256 (2009) [CrossRef] .

15. S. P. Stark, T. Steinmetz, R. A. Probst, H. Hundertmark, T. Wilken, T. W. Hänsch, Th. Udem, P. St. J. Russell, and R. Holzwarth, “14 GHz visible supercontinuum generation: calibration sources for astronomical spectrographs,” Opt. Express **19**, 15690–15695 (2011) [CrossRef] [PubMed] .

16. T. Wilken, G. Lo Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. Gonzalez Hernandez, R. Rebolo, T. W. Haensch, Th. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature **485**, 611–614 (2012) [CrossRef] [PubMed] .

17. F. Quinlan, G. Ycas, S. Osterman, and S. A. Diddams, “A 12.5 GHz-spaced optical frequency comb spanning >400 nm for near-infrared astronomical spectrograph calibration,” Rev. Sci. Instrum. **81**, 063105 (2010) [CrossRef] [PubMed] .

18. A. J. Benedick, G. Chang, J. R. Birge, L.-J. Chen, A. G. Glenday, C.-H. Li, D. F. Phillips, A. Szentgyorgyi, S. Korzennik, G. Furesz, R. L. Walsworth, and F. X. Kärtner, “Visible wavelength astro-comb,” Opt. Express **18**, 19175–19184 (2010) [CrossRef] [PubMed] .

20. D. F. Phillips, A. G. Glenday, C.-H. Li, C. Cramer, G. Furesz, G. Chang, A. J. Benedick, L.-J. Chen, F. X. Kärtner, S. Korzennik, D. Sasselov, A. Szentgyorgyi, and R. L. Walsworth, “Calibration of an astrophysical spectrograph below 1 m/s using a laser frequency comb,” Opt. Express **20**, 13711–13726 (2012) [CrossRef] [PubMed] .

*m*> 2, for which the cavity transmission phase causes asymmetries. We also explain, why effects such as Raman gain, self-steepening, and guided acoustic-wave Brillouin scattering (GAWBS) can be neglected in our description. Besides, we provide plots that allow scaling to other systems in order to estimate the required side-mode suppression after filtering.

## 2. Theoretical model

*m*= 2, with only one side-mode between two principal modes. Under these conditions the side-modes are essentially symmetric by construction. We use the time domain description with a train of alternating strong and weak pulses. The weak pulses are scaled by a factor

*h*≤ 1 in field amplitude (see Fig. 1). In the frequency domain, the initial side-mode suppression in terms of power for

*m*= 2 is then given by

*R*(

_{i}*h*) ≡ (1 +

*h*)

^{2}/(1 −

*h*)

^{2}. A train of identical pulses with

*h*= 1 corresponds to infinite side-mode suppression

*R*(

_{i}*h*= 1) = ∞, whereas a finite suppression is obtained for

*h*< 1. We consider Gaussian pulses with a normalized electric field amplitude before nonlinear propagation (

*z*= 0) given by [24]: where

*T*=

*t*−

*z/v*is the time in a retarded reference frame, with the time

_{g}*t*, the propagation length

*z*, and the group velocity

*v*.

_{g}*C*is the temporal frequency-chirp, and

*T*of the optical power of the pulse. The propagation length

_{FWHM}*z*determines the intensity-dependent nonlinear phase

*ϕ*: with the nonlinear length

_{NL}*L*= (

_{NL}*γP*

_{0})

^{−1}, the peak power of the pulse

*P*

_{0}, and

*γ*=

*n*

_{2}

*ω*as defined in [24] with the nonlinear refractive index

_{c}/cA_{eff}*n*

_{2}, the optical carrier frequency

*ω*and the effective beam cross section

_{c}*A*. A typical value of

_{eff}*γ*for the experimental conditions described in Section 3 is 1 W

^{−1}m

^{−1}[15

15. S. P. Stark, T. Steinmetz, R. A. Probst, H. Hundertmark, T. Wilken, T. W. Hänsch, Th. Udem, P. St. J. Russell, and R. Holzwarth, “14 GHz visible supercontinuum generation: calibration sources for astronomical spectrographs,” Opt. Express **19**, 15690–15695 (2011) [CrossRef] [PubMed] .

*T*can be taken into account via the shift theorem. Its nonlinear shift is smaller by

_{r}*h*

^{2}, which can be included in the effective propagation distance. To calculate the power spectral density

*S*(

*ω*) at frequency

*ω*after SPM, it is sufficient to consider a single repetition of the optical pulse sequence, here a double pulse:

*S*(

*ω*) is evaluated at frequencies

*ω*= 2

*nω*, and for the side-modes at frequencies

_{r}*ω*= (2

*n*+1)

*ω*, with the repetition rate

_{r}*ω*= 2

_{r}*π/T*and integer mode number

_{r}*n*. For the modeling we set the carrier frequency

*ω*to zero as it only shifts the resulting spectrum.

_{c}*R*(

_{i}*h*) is set to 74 dB (⇒ 1 −

*h*= 4 × 10

^{−4}), the pulse duration to 100 fs FWHM with a repetition rate of

*ω*= 2

_{r}*π*× 9 GHz. The pulse duration is assumed to be initially transform limited (

*C*= 0), and the propagation length

*z*is chosen to be 44

*L*. The resulting spectra before and after nonlinear propagation are shown in Fig. 2. Besides the obvious degradation of side-mode suppression, it can be seen, that SPM imprints a structure on the spectra, which is more pronounced for the principal mode spectrum than for the side-mode spectrum. Figure 3 shows the resulting side-mode suppression for different values of the initial suppression under otherwise identical conditions. For the interesting case of well suppressed initial side-modes, we find that the amount and spectral structure of the side-mode amplification is independent of the initial suppression. Under these conditions the side-mode gain due to SPM is between 35 and 46 dB. For poorly suppressed initial side-modes the gain spectrum changes considerably, and we can observe the optical power to be transferred back to the principal modes.

_{NL}*m*= 6 and

*m*= 20. For

*m*> 2 the side-modes acquire asymmetric phase shifts upon transmission through the cavity which turn into asymmetric powers after SPM [13

13. G. Chang, C.-H. Li, D. F. Phillips, R. L. Walsworth, and F. X. Kaertner, “Toward a broadband astro-comb: effects of nonlinear spectral broadening in optical fibers,” Opt. Express **18**, 12736–12747 (2010) [CrossRef] [PubMed] .

*m*= 2, i.e. Eq. (3) with equal initial suppression. The approximation through

*m*= 2 improves with larger initial side-mode suppression as the asymmetry of the side-mode gain decreases.

*f*then is given by:

_{RMS}*z*and small initial chirps

*C*, Δ

*f*becomes linear in both

_{RMS}*z/L*and

_{NL}*C*:

*C*= +5,

*C*= 0 and

*C*= −5. Notice that in Fig. 4, the pulses have identical pulse durations

*T*= 100 fs but different transform limits, depending on the value of the initial chirp

_{FWHM}*C. C*= ±5 corresponds to 5 times the transform limited time-bandwidth product, or in this case to a transform limit of 20 fs FWHM. We see, that the linear approximation given by Eq. (5) is quite good except for the first few nonlinear lengths.

*A*. Similarly to Δ

_{avg}*f*, its evolution with

_{RMS}*z/L*can be obtained analytically (see appendix), and is found to be given by:

_{NL}*z*in terms of

*L*. Later in this section, we show how to estimate

_{NL}*z*in terms of

*L*experimentally from the observed spectral broadening via Eq. (5). In contrast to the average side-mode suppression, the lowest side-mode suppression within the −20 dB-bandwidth depends on the initial chirp. For initially transform limited pulses it is never more than 10 dB below the average suppression, see Fig. 4(b).

_{NL}*L*. As

_{NL}*A*is independent of pulse duration and chirp, the evolution of

_{avg}*A*with

_{avg}*z*in terms of

*L*is expected to be unaffected by dispersion. The RMS-bandwidth, however, will no longer increase linearly while the pulse duration changes, as its slope scales with the inverse of

_{NL}*T*, see Eq. (5). So, if dispersion monotonically stretches the pulses, the broadening will be smaller than the one given by Eq. (5), if

_{FWHM}*T*is the initial FWHM pulse duration. On the other hand, if the pulses are compressed during propagation, the broadening will be larger. Spectral broadening usually only takes place in the latter scenario, as only then can nonlinearities be driven efficiently at moderate pulse energies. Pulse compression during propagation usually takes place in the anomalous dispersion regime, where the interplay of nonlinearity and dispersion can compress even initially transform limited pulses.

_{FWHM}*L*changes during propagation due to changing peak power. Therefore, the propagation length in terms of

_{NL}*L*is now given by

_{NL}*z/L*in Eqs. (5) and (6). The fiber dispersion is quantified with respect to the initial nonlinear length. We assume a group velocity dispersion of +25 fs

_{NL}^{2}per initial

*L*for the case of normal dispersion, and −25 fs

_{NL}^{2}for anomalous dispersion. In the latter case, the pulses evolve into 12

*order solitons. The evolution into a second-order soliton is obtained by assuming −1000 fs*

^{th}^{2}per initial

*L*. We also simulated the propagation of an initial first-order soliton assuming −3220 fs

_{NL}^{2}per initial

*L*and an initial 100 fs sech

_{NL}^{2}-pulse shape. Figure 6 shows the evolution of the RMS-bandwidth and average suppression for these regimes. We find, that the side-mode amplification follows the curve for the non-dispersive propagation within ±5 dB in all cases. The deviation is probably due to the deformed pulse shape. The evolution of the RMS-bandwidth can generally be well understood by the above discussion, as long as soliton formation has not yet occurred. However, as soon as a soliton has formed, spectral broadening essentially stops, but side-mode amplification continues. In the case of higher order solitons, we also observe oscillations of the RMS-bandwidth. These oscillations are not likely to occur in experiments, as higher-order solitons usually decay into fundamental solitons in more realistic scenarios [24].

*L*is obtained from the observed spectral broadening via Eq. (5). In combination with Eq. (6) this gives an upper limit on the average side-mode amplification. Based on the results for the non-dispersive scenario, the lowest side-mode suppression is less than a factor of 10 below the average suppression. A worst-case estimation for the side-mode amplification

_{NL}*A*as a function of initial pulse duration

*T*and final RMS-bandwidth Δ

_{FWHM}*f*can then be given as:

_{RMS}13. G. Chang, C.-H. Li, D. F. Phillips, R. L. Walsworth, and F. X. Kaertner, “Toward a broadband astro-comb: effects of nonlinear spectral broadening in optical fibers,” Opt. Express **18**, 12736–12747 (2010) [CrossRef] [PubMed] .

26. M. S. Kang, A. Nazarkin, A. Brenn, and P. St. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. **5**, 276–280 (2009) [CrossRef] .

26. M. S. Kang, A. Nazarkin, A. Brenn, and P. St. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. **5**, 276–280 (2009) [CrossRef] .

*P*

^{2}

*(*

^{ω}*t*) = 2

*ε*

_{0}

*χ*

^{(2)}

*E*

^{2}(

*t*), which mostly generates sum frequencies in this context, produces a frequency comb with the same mode spacing but doubles the carrier-envelope frequency. To examine what happens to the side-mode suppression in this case, we again assume an alternating pulse sequence with

*m*= 2. Assuming an undepleted pump wave, SHG scales with the square of the fundamental power such that the side-mode amplification is given by

*R*(

_{i}*h*)/

*R*(

_{i}*h*

^{2}) = 4 −

*𝒪*[(1 −

*h*)

^{2}]. The side-modes are therefore amplified by 6 dB in power upon SHG, provided that the side-modes are sufficiently suppressed (

*h*→ 1). The same is true for any other value of

*m*.

## 3. Experimental setup

14. C.-H. Li, A. G. Glenday, A. J. Benedick, G. Chang, L.-J. Chen, C. Cramer, P. Fendel, G. Furesz, F. X. Kärtner, S. Korzennik, D. F. Phillips, D. Sasselov, A. Szentgyorgyi, and R. L. Walsworth, “In-situ determination of astro-comb calibrator lines to better than 10 cm s^{−1},” Opt. Express **18**, 13239–13249 (2010) [CrossRef] [PubMed] .

15. S. P. Stark, T. Steinmetz, R. A. Probst, H. Hundertmark, T. Wilken, T. W. Hänsch, Th. Udem, P. St. J. Russell, and R. Holzwarth, “14 GHz visible supercontinuum generation: calibration sources for astronomical spectrographs,” Opt. Express **19**, 15690–15695 (2011) [CrossRef] [PubMed] .

27. L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, and T. W. Hänsch, “A compact grating-stabilized diode-laser system for atomic physics,” Opt. Commun. **117**, 541–549 (1995) [CrossRef] .

28. Z. F. Fan, P. J. S. Heim, and M. Dagenais, “Highly coherent RF signal generation by heterodyne optical phase locking of external cavity semiconductor lasers,” IEEE Photonics Technol. Lett. **10**, 719–721 (1998) [CrossRef] .

_{3}-crystal. The resulting 1.5 mW of green light has a strongly astigmatic beam profile, which is compensated by a tilted curved mirror and a lens. Finally, both the astro-comb and the green cw light are coupled through the same single-mode fiber for mode matching with strong damping in the infrared (IR) and then focused onto a fast photodiode.

## 4. Experimental results

12. T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, and Th. Udem, “Fabry-Perot filter cavities for wide-spaced frequency combs with large spectral bandwidth,” Appl. Phys. B **96**, 251–256 (2009) [CrossRef] .

29. T. Wilken, C. Lovis, A. Manescau, T. Steinmetz, L. Pasquini, G. Lo Curto, T. W. Hänsch, R. Holzwarth, and Th. Udem, “High-precision calibration of spectrographs,” Mon. Not. R. Astron. Soc. **405**, L16–L20 (2010) [CrossRef] .

*T*=100 fs plus 6 dB resulting from the SHG. Given the initial side-mode suppression of 38 dB this actually means, that some of the approximations of Eq. (7) break down, and the side-modes can become stronger than the principal modes (see Fig. 5).

_{FWHM}16. T. Wilken, G. Lo Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. Gonzalez Hernandez, R. Rebolo, T. W. Haensch, Th. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature **485**, 611–614 (2012) [CrossRef] [PubMed] .

*T*=100 fs, which predicts a suppression of at least 28 dB. Since this suppression would be insufficient for the most challenging future applications of the astro-comb, we added a fourth FPC for the measurements presented in [16

_{FWHM}16. T. Wilken, G. Lo Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. Gonzalez Hernandez, R. Rebolo, T. W. Haensch, Th. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature **485**, 611–614 (2012) [CrossRef] [PubMed] .

30. H.-P. Doerr, T. Steinmetz, R. Holzwarth, T. Kentischer, and W. Schmidt, “Laser frequency comb system for absolute calibration of the VTT echelle spectrograph,” Solar Phys. **280**, 663–670 (2012) [CrossRef] .

*T*=100 fs we expect a side-mode suppression of at least 16 dB. The suppression was measured at 532 nm, 521 nm and 515 nm, as shown in Fig. 12. Again, we observed spontaneous changes of the side-mode suppression, but in this case on a time scale of hours, as seen on the data points at 532 nm. A significant polarization dependence was also observed. By rotating the half-wave plate in front of the PCF we could collect a number of different data points within a short time. As can be seen from Fig. 12, none of the data points exceeded the expected upper limit.

_{FWHM}31. H.-P. Doerr, T. J. Kentischer, T. Steinmetz, R. A. Probst, M. Franz, R. Holzwarth, Th. Udem, T. W. Hänsch, and W. Schmidt, “Performance of a laser frequency comb calibration system with a high-resolution solar echelle spectrograph,” Proc. SPIE **8450**, 84501G (2012) [CrossRef] .

31. H.-P. Doerr, T. J. Kentischer, T. Steinmetz, R. A. Probst, M. Franz, R. Holzwarth, Th. Udem, T. W. Hänsch, and W. Schmidt, “Performance of a laser frequency comb calibration system with a high-resolution solar echelle spectrograph,” Proc. SPIE **8450**, 84501G (2012) [CrossRef] .

## 5. Conclusion

## 6. Appendix

*B*≡ 1 −

*h*and compute the leading order in

*B*. The filter ratio is set to

*m*= 2

*,*and the carrier frequency to

*ω*= 0. As explained in Section 2, the result in terms of spectral broadening and side-mode amplification is approximately independent of that choice. The initial side-mode suppression is given by

_{c}*R*= (1 +

_{i}*h*)

^{2}/(1 −

*h*)

^{2}≈ 4/

*B*

^{2}. Using Eq. (3), the powers of the

*n*principal mode and of the

^{th}*n*side-mode are given by

^{th}*P*=

_{n}*S*(2

*nω*) and

_{r}*S*=

_{n}*S*((2

*n*+ 1)

*ω*), respectively. To describe the average side-mode suppression

_{r}*R*after nonlinear propagation, we average the relative side-mode power

_{avg}*S*, weighted with the corresponding principal mode power

_{n}/P_{n}*P*, and invert the result. This yields

_{n}*A*is then given by:

_{avg}*f*can be expressed as:

_{RMS}*A*and Δ

_{avg}*f*are determined by the three sums

_{RMS}*h*

^{2}≈ 1 − 2

*B*, and

*U*(

*z*,

*T*) as defined by Eqs. (1) and (2), the power of the

*n*principal mode becomes:

^{th}*n*has been replaced by an approximating integral leading to a

*δ*-function. Expanding the term in square brackets yields 4−4

*B*+(1−

*A*(

*T*)

^{2})

*B*

^{2}+

*O*(

*B*

^{3}), which is independent of

*B*in first order (undepleted principal modes). With the remaining integral over

*A*(

*T*)

^{2})

*B*

^{2}+

*O*(

*B*

^{3}). The remaining integral over

*δ*″(

*x*) denotes the second derivative of the Dirac delta function:

## Acknowledgments

## References and links

1. | Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature |

2. | M. Mayor and D. Queloz, “A Jupiter-mass companion to a solar-type star,” Nature |

3. | J. Schneider, “The extrasolar planets encyclopaedia,” http://exoplanet.eu/catalog.php. |

4. | J. Liske, A. Grazian, E. Vanzella, M. Dessauges, M. Viel, L. Pasquini, M. Haehnelt, S. Cristiani, F. Pepe, G. Avila, P. Bonifacio, F. Bouchy, H. Dekker, B. Delabre, S. D’Odorico, V. D’Odorico, S. Levshakov, C. Lovis, M. Mayor, P. Molaro, L. Moscardini, M. T. Murphy, D. Queloz, P. Shaver, S. Udry, T. Wiklind, and S. Zucker, “Cosmic dynamics in the era of extremely large telescopes,” Mon. Not. R. Astron. Soc. |

5. | M. T. Murphy, Th. Udem, R. Holzwarth, A. Sizmann, L. Pasquini, C. Araujo-Hauck, H. Dekker, S. D’Odorico, M. Fischer, T. W. Hänsch, and A. Manescau, “High-precision wavelength calibration of astronomical spectrographs with laser frequency combs,” Mon. Not. R. Astron. Soc. |

6. | A. Bartels, D. Heinecke, and S. A. Diddams, “10-GHz self-referenced optical frequency comb,” Science |

7. | J. J. McFerran, L. Nenadovic, W. C. Swann, J. B. Schlager, and N. R. Newbury, “A passively mode-locked fiber laser at 1.54 |

8. | H.-W. Chen, G. Chang, S. Xu, Z. Yang, and F. X. Kärtner, “3 GHz, fundamentally mode-locked, femtosecond Yb-fiber laser,” Opt. Lett. |

9. | 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 Th. Udem, “Laser frequency combs for astronomical observations,” Science |

10. | C.-H. Li, A. J. Benedick, P. Fendel, A. G. Glenday, F. X. Kärtner, D. F. Phillips, D. Sasselov, A. Szentgyorgyi, and R. L. Walsworth, “A laser frequency comb that enables radial velocity measurements with a precision of 1 cm s |

11. | D. A. Braje, M. S. Kirchner, S. Osterman, T. Fortier, and S. A. Diddams, “Astronomical spectrograph calibration with broad-spectrum frequency combs,” Eur. Phys. J. D |

12. | T. Steinmetz, T. Wilken, C. Araujo-Hauck, R. Holzwarth, T. W. Hänsch, and Th. Udem, “Fabry-Perot filter cavities for wide-spaced frequency combs with large spectral bandwidth,” Appl. Phys. B |

13. | G. Chang, C.-H. Li, D. F. Phillips, R. L. Walsworth, and F. X. Kaertner, “Toward a broadband astro-comb: effects of nonlinear spectral broadening in optical fibers,” Opt. Express |

14. | C.-H. Li, A. G. Glenday, A. J. Benedick, G. Chang, L.-J. Chen, C. Cramer, P. Fendel, G. Furesz, F. X. Kärtner, S. Korzennik, D. F. Phillips, D. Sasselov, A. Szentgyorgyi, and R. L. Walsworth, “In-situ determination of astro-comb calibrator lines to better than 10 cm s |

15. | S. P. Stark, T. Steinmetz, R. A. Probst, H. Hundertmark, T. Wilken, T. W. Hänsch, Th. Udem, P. St. J. Russell, and R. Holzwarth, “14 GHz visible supercontinuum generation: calibration sources for astronomical spectrographs,” Opt. Express |

16. | T. Wilken, G. Lo Curto, R. A. Probst, T. Steinmetz, A. Manescau, L. Pasquini, J. I. Gonzalez Hernandez, R. Rebolo, T. W. Haensch, Th. Udem, and R. Holzwarth, “A spectrograph for exoplanet observations calibrated at the centimetre-per-second level,” Nature |

17. | F. Quinlan, G. Ycas, S. Osterman, and S. A. Diddams, “A 12.5 GHz-spaced optical frequency comb spanning >400 nm for near-infrared astronomical spectrograph calibration,” Rev. Sci. Instrum. |

18. | A. J. Benedick, G. Chang, J. R. Birge, L.-J. Chen, A. G. Glenday, C.-H. Li, D. F. Phillips, A. Szentgyorgyi, S. Korzennik, G. Furesz, R. L. Walsworth, and F. X. Kärtner, “Visible wavelength astro-comb,” Opt. Express |

19. | M. T. Murphy, C. R. Locke, P. S. Light, A. N. Luiten, and J. S. Lawrence, “Laser frequency comb techniques for precise astronomical spectroscopy,” Mon. Not. R. Astron. Soc. |

20. | D. F. Phillips, A. G. Glenday, C.-H. Li, C. Cramer, G. Furesz, G. Chang, A. J. Benedick, L.-J. Chen, F. X. Kärtner, S. Korzennik, D. Sasselov, A. Szentgyorgyi, and R. L. Walsworth, “Calibration of an astrophysical spectrograph below 1 m/s using a laser frequency comb,” Opt. Express |

21. | G. G. Ycas, F. Quinlan, S. A. Diddams, S. Osterman, S. Mahadevan, S. Redman, R. Terrien, L. Ramsey, C. F. Bender, B. Botzer, and S. Sigurdsson, “Demonstration of on-sky calibration of astronomical spectra using a 25 GHz near-IR laser frequency comb,” Opt. Express |

22. | T. Wilken, R. Probst, T. W. Hänsch, Th. Udem, T. Steinmetz, R. Holzwarth, A. Manescau, G. L. Curto, L. Pasquini, S. Stark, H. Hundertmark, and P. St. J. Russell, “Suppressed mode recovery in nonlinear fibers of a Fabry-Perot-filtered frequency comb,” in “CLEO:2011 - Laser Applications to Photonic Applications,” (Optical Society of America, 2011), p. CWQ2. |

23. | G. Chang, C.-H. Li, D. F. Phillips, A. Szentgyorgyi, R. L. Walsworth, and F. X. Kärtner, “Optimization of filtering schemes for broadband astro-combs,” Opt. Express |

24. | G. P. Agrawal, |

25. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

26. | M. S. Kang, A. Nazarkin, A. Brenn, and P. St. J. Russell, “Tightly trapped acoustic phonons in photonic crystal fibres as highly nonlinear artificial Raman oscillators,” Nat. Phys. |

27. | L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, and T. W. Hänsch, “A compact grating-stabilized diode-laser system for atomic physics,” Opt. Commun. |

28. | Z. F. Fan, P. J. S. Heim, and M. Dagenais, “Highly coherent RF signal generation by heterodyne optical phase locking of external cavity semiconductor lasers,” IEEE Photonics Technol. Lett. |

29. | T. Wilken, C. Lovis, A. Manescau, T. Steinmetz, L. Pasquini, G. Lo Curto, T. W. Hänsch, R. Holzwarth, and Th. Udem, “High-precision calibration of spectrographs,” Mon. Not. R. Astron. Soc. |

30. | H.-P. Doerr, T. Steinmetz, R. Holzwarth, T. Kentischer, and W. Schmidt, “Laser frequency comb system for absolute calibration of the VTT echelle spectrograph,” Solar Phys. |

31. | H.-P. Doerr, T. J. Kentischer, T. Steinmetz, R. A. Probst, M. Franz, R. Holzwarth, Th. Udem, T. W. Hänsch, and W. Schmidt, “Performance of a laser frequency comb calibration system with a high-resolution solar echelle spectrograph,” Proc. SPIE |

**OCIS Codes**

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(120.6200) Instrumentation, measurement, and metrology : Spectrometers and spectroscopic instrumentation

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(300.6310) Spectroscopy : Spectroscopy, heterodyne

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 6, 2013

Revised Manuscript: April 16, 2013

Manuscript Accepted: April 20, 2013

Published: May 6, 2013

**Citation**

R. A. Probst, T. Steinmetz, T. Wilken, H. Hundertmark, S. P. Stark, G. K. L. Wong, P. St. J. Russell, T. W. Hänsch, R. Holzwarth, and Th. Udem, "Nonlinear amplification of side-modes in frequency combs," Opt. Express **21**, 11670-11687 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-11670

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