Nonlinear amplification of side-modes in frequency combs

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; Th. Udem

doi:10.1364/OE.21.011670

1. Introduction

Since their introduction more than 10 years ago, frequency combs [1] have become standard devices for measuring the frequency of lasers in high resolution spectroscopy or optical clocks. While this method represents the solution to essentially all tasks involving the frequency measurement of coherent laser radiation, there are important applications where a temporally incoherent source needs to be measured as accurately as possible. Precision spectroscopy in astronomy for example plays a key role in finding extra-solar planets by detecting tiny periodic Doppler shifts due to the recoiling quiver motion of their host stars. This technique has enabled the first and so far the most numerous discoveries of extra-solar planets [2, 3]. However, the discovery of Earth-like planets with this technique has been inhibited by the lack of sufficient accuracy. Direct observation of the accelerated expansion of the universe, to help understanding the nature of dark energy, is another example that requires an even higher accuracy [4].

The accuracy of the best astronomical spectrographs used for these purposes has been limited by the available calibration sources, mostly spectral lamps and iodine absorption cells. This limitation could be overcome by the use of frequency combs as regular calibration grids [5], if the modes can be resolved by the spectrograph. For a realistic case this usually requires a mode spacing of 10 to 30 GHz. Although Ti:sapphire lasers have been demonstrated with a 10 GHz mode spacing [6], these lasers are not yet in a state where they could be operated autonomously for an extended period at a remote telescope. Mode-locked fiber lasers have the capability to do so, but the largest mode spacings demonstrated so far are 2 to 3 GHz [7, 8].

Therefore, the prevailing method to obtain an astronomical wavelength calibrator from a frequency comb is to filter its spectrum by the almost periodic transmission function of a Fabry-Pérot cavity (FPC) [9–12]. The FPC’s free spectral range (FSR) is set to an integer multiple m of the mode spacing of the frequency comb and stabilized to it. This suppresses all but every m^th 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 F, the suppression factor of the closest mode is given approximately by (2F/m)². Limited side-mode suppression can lead to calibration errors, if the side-modes are not spectrally resolved [11, 13, 14]. In particular a suppression ratio that is asymmetric around the principal modes can be detrimental to the calibration accuracy [11]. This can be caused by a cavity-comb spectral walk-off due to the finite cavity dispersion. Another source of asymmetry can be the excitation of higher order transversal modes due to a less than perfect spatial mode matching to the cavity. By picking the proper mirror curvatures this effect can be minimized [12].

Fig. 1 (a) Frequency domain: A Fabry-Pérot cavity is used as a spectral filter. Its free spectral range is an integer multiple (here m = 4) of the initial mode spacing ω_r = 2π/T_r. (b) The finite finesse of the cavity leads to unwanted side-modes to each principal mode. (c) In the time domain the pulse stored inside the cavity gets replenished by the next pulse from the initial pulse train only every m^th round trip. The finite finesse leads to an exponential decay of the pulse energy between these events with a normalized electric field reduction h. The resulting amplitude modulation causes the side bands to appear.

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The spectrum of the frequency comb should ideally cover the full bandwidth of the spectrograph, which can be as large as an octave. The spectrum emitted by the laser that generates the frequency comb usually does not match the spectrograph’s bandwidth and thus needs to be manipulated. While the emission range of frequency combs usually is in the far red to infrared regime, high precision spectrographs typically cover the short to mid-wavelength range of the visible spectrum, where most stars have a large selection of Fraunhofer lines. Frequency conversion processes such as spectral broadening in specially designed photonic crystal fibers (PCFs) [15] and second-harmonic generation (SHG) can be used to tailor the spectrum as desired [16, 17]. Both processes are most conveniently performed after filtering the comb to the full mode spacing. This is because filtering a broadened spectrum is technically difficult due to intra-cavity dispersion of FPCs. Additionally, after frequency conversion, the losses due to filtering can normally not be compensated by a subsequent amplifier. Especially when having to broaden the spectrum after frequency doubling, high pulse energies are required. Alternative concepts have been demonstrated where these drawbacks can be tolerated [18–20].

It has been shown both experimentally [16, 17, 21, 22] and in numerical simulations [13, 23], that nonlinear spectral broadening reduces side-mode suppression. The side-modes are thus amplified relative to their adjacent principal modes, which we hereafter refer to as side-mode amplification. This amplification may be asymmetric for the lower and upper side-mode, which causes calibration errors as the spectrograph does not resolve them. The asymmetry depends on the dispersion of the filter cavities and fibers. Chang et al. [13, 23] have investigated this effect by modeling in detail dispersion and Raman related asymmetric side-mode suppression.

In this paper we ignore these effects and instead find regimes where the reamplification of side-modes is below a threshold for which asymmetries can be neglected. We believe that it would be difficult in practice to control asymmetry with sufficient accuracy. In fact we assume maximum asymmetric side-modes to estimate the calibration errors. Our model takes into account self-phase modulation (SPM) and SHG only, and neglects effects that generate side-mode asymmetries. This approximation yields compact expressions for side-mode amplification and spectral broadening, that are valid in the regimes of interest. These regimes assume sufficient initial side-mode suppression and a nonlinear optical fiber limited in length to where spectral broadening comes to a halt (to avoid solitonic effects). We verify that our approximations are useful by comparing the results with experimental data and with full fetched split-step Fourier method, that takes dispersion into account, as well as testing with 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

Our model for side-mode amplification incorporates the main physical mechanism for spectral broadening in nonlinear fibers, which is SPM. To keep our theoretical treatment as simple as possible, we neglect the Raman effect, dispersion and self-steepening, and examine the case of 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]:

U (z = 0, T) = exp [- (1 + i C) \frac{T^{2}}{2 T_{0}^{2}}]

where T = t − z/v_g is the time in a retarded reference frame, with the time t, the propagation length z, and the group velocity v_g. C is the temporal frequency-chirp, and

T_{0} = T_{F W H M} / [2 \sqrt{ln (2)}]

, with the full width at half maximum (FWHM) pulse duration T_FWHM of the optical power of the pulse. The propagation length z determines the intensity-dependent nonlinear phase ϕ_NL:

U (z, T) = U (0, T) exp (i ϕ_{N L}), ϕ_{N L} = {| U (0, T) |}^{2} \frac{z}{L_{N L}}

with the nonlinear length L_NL = (γP₀)⁻¹, the peak power of the pulse P₀, and γ = n₂ω_c/cA_eff as defined in [24] with the nonlinear refractive index n₂, the optical carrier frequency ω_c and the effective beam cross section A_eff. A typical value of γ for the experimental conditions described in Section 3 is 1 W⁻¹ m⁻¹[15], such that for 50 W peak power pulses the nonlinear length is 2 cm. The spectral envelope of such an alternating pulse train after SPM is obtained by Fourier transformation, where the time shift of the weaker pulse of half the repetition period T_r can be taken into account via the shift theorem. Its nonlinear shift is smaller by h², 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) = {| \int_{- \infty}^{+ \infty} [U (z, T) + h U (h^{2} z, T) exp (i ω \frac{T_{r}}{2})] exp [i (ω - ω_{c}) T] d T |}^{2}

To compute this integral with a strongly oscillatory integrand numerically, we found the Runge-Kutta-method with adaptive step-size control [25] or fast Fourier transformation [25] to be effective. The latter proved to be faster and more precise. For the principal modes, S(ω) is evaluated at frequencies ω = 2nω_r, and for the side-modes at frequencies ω = (2n+1)ω_r, with the repetition rate ω_r = 2π/T_r and integer mode number n. For the modeling we set the carrier frequency ω_c to zero as it only shifts the resulting spectrum.

To facilitate comparison with our experimental results presented in Section 4, we demonstrate the behavior for the following parameters: The initial suppression 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_r = 2π × 9 GHz. The pulse duration is assumed to be initially transform limited (C = 0), and the propagation length z is chosen to be 44L_NL. 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.

Fig. 2 Spectral envelope of the principal modes (black line) and interleaved side-modes (red line), for a filter ratio of m = 2 and an initial suppression of 74 dB. The optical pulses are initially transform limited with 100 fs FWHM pulse duration and Gaussian shape. (a) Before nonlinear propagation. (b) After nonlinear propagation over a distance of 44 times the nonlinear length.

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Fig. 3 Side-mode suppression versus optical frequency after propagation over 44 times the nonlinear length for different values of the initial side-mode suppression. The initial suppression is written at each curve, and the filter ratio is m = 2. The optical pulses have a 100 fs FWHM pulse duration with a Gaussian envelope and no initial chirp.

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The calculations were repeated for the filter ratios 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]. It was found that as long as the side-modes stay considerably weaker than the principal modes, the final powers of all side-modes can be very well approximated by the case 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.

The investigation of the (usually unwanted) side-mode gain has to be complemented by the (usually wanted) spectral broadening, because both effects appear simultaneously. We quantify the width of the rather structured spectra by their root-mean-square-bandwidth (RMS-bandwidth), which is computed as the standard deviation of the power spectrum, see Eq. (9) in the appendix. For the side-mode amplification, we average the power-ratios of all side-modes relative to their adjacent principal modes, weighted by the power of each principal mode. The increase of this value measures the average side-mode amplification, and the inverse of this value is used to quantify the average side-mode suppression. Additionally, we identified the single side-mode with lowest side-mode suppression within the −20 dB-bandwidth of the overall spectral envelope. This can serve as an estimation of the worst-case side-mode suppression.

As demonstrated in the appendix, the evolution of the RMS-bandwidth with propagation length can be calculated analytically in the limit of high side-mode suppression. The evolution of the RMS-bandwidth Δf_RMS then is given by:

2 π Δ f_{R M S} = \frac{2 \sqrt{ln (2)}}{T_{F W H M}} \sqrt{\frac{1}{2} + \frac{C^{2}}{2} + \frac{C}{\sqrt{2}} \frac{z}{L_{N L}} + \frac{2}{3 \sqrt{3}} {(\frac{z}{L_{N L}})}^{2}}

For large propagation lengths z and small initial chirps C, Δf_RMS becomes linear in both z/L_NL and C:

2 π Δ f_{R M S} \approx \frac{1}{T_{F W H M}} \frac{2 \sqrt{2 ln (2)}}{\sqrt{3 \sqrt{3}}} (\frac{3}{4} \sqrt{\frac{3}{2}} C + \frac{z}{L_{N L}})

The last two equations are compared in Fig. 4(a) for C = +5, C = 0 and C = −5. Notice that in Fig. 4, the pulses have identical pulse durations T_FWHM = 100 fs but different transform limits, depending on the value of the initial chirp 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.

Fig. 4 (a) Evolution of the RMS-bandwidth as a function of the propagation length z for Gaussian pulses with 100 fs FWHM pulse duration, 74 dB initial side-mode suppression and different initial chirps C. Red dashed line: Approximation given by Eq. (5). (b) Evolution of the average side-mode suppression (independent of chirp and pulse duration) and the lowest side-mode suppression within the −20 dB-bandwidth for initially unchirped pulses.

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The average side-mode amplification is quantified by the amplification factor A_avg. Similarly to Δf_RMS, its evolution with z/L_NL can be obtained analytically (see appendix), and is found to be given by:

A_{avg} = 1 + \frac{4}{\sqrt{3}} {(\frac{z}{L_{N L}})}^{2}

This relation is an important conclusion of the paper. Remarkably, it is found to be valid independently of pulse duration and chirp. As explained below, it is also independent of the initial side-mode suppression and fiber dispersion within reasonable limits. The average side-mode amplification can thus be obtained by solely determining the propagation length z in terms of L_NL. Later in this section, we show how to estimate z in terms of L_NL 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).

Equations (4)–(6) hold as long as the side-modes do not exceed the principal modes in power. This can be seen from Fig. 5, where the Eqs. (5) and (6) are compared with results from Eq. (3) for different initial suppressions. Significant deviations are observed in Fig. 5 as soon as the side-mode suppression approaches the 0 dB-level. As only the case of well-suppressed side-modes is suitable for practical applications, this is not a substantial limitation to the usefulness of Eqs. (4)–(6).

Fig. 5 (a) RMS-bandwidth versus propagation length for different initial suppressions. For initial suppressions ≥54 dB the curves are virtually identical. The dashed line is the linear approximation given by Eq. (5). (b) Average side-mode suppression versus propagation length for different initial suppressions. Here, the dashed lines represent the approximation for high side-mode suppression given by Eq. (6). In all cases, the pulses are assumed to have a 100 fs FWHM duration, Gaussian shape, and no initial chirp.

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We now show how dispersion, which has been neglected so far, can be included in the above considerations. The main effect of dispersion is to alter the pulse duration and chirp. The resulting modulation of the peak power merely changes the nonlinear length L_NL. As A_avg is independent of pulse duration and chirp, the evolution of A_avg with z in terms of L_NL 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 T_FWHM, see Eq. (5). So, if dispersion monotonically stretches the pulses, the broadening will be smaller than the one given by Eq. (5), if T_FWHM 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.

In order to verify these simplified arguments about the influence of dispersion, we use the split-step Fourier method [24], that can precisely describe nonlinear pulse propagation under dispersion. In this case the value of L_NL changes during propagation due to changing peak power. Therefore, the propagation length in terms of L_NL is now given by $\int_{0}^{z} [1 / L_{N L} (z^{'})] d z^{'}$ , which replaces z/L_NL 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 fs² per initial L_NL for the case of normal dispersion, and −25 fs² for anomalous dispersion. In the latter case, the pulses evolve into 12^th order solitons. The evolution into a second-order soliton is obtained by assuming −1000 fs² per initial L_NL. We also simulated the propagation of an initial first-order soliton assuming −3220 fs² per initial L_NL and an initial 100 fs sech²-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].

Fig. 6 Evolution of (a) the RMS-bandwidth and (b) the average side-mode suppression during nonlinear propagation for initially unchirped 100 fs-pulses in different dispersion regimes. The curve for normal dispersion assumes +25 fs² per initial L_NL. In the case of anomalous dispersion solitons of various orders are formed depending on the amount of dispersion. The pulses are launched with a Gaussian shape except for the first order soliton, which starts out as a sech² pulse.

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The nonlinear fiber length and optical power should therefore be dimensioned in a way, that soliton formation, marked by a fast increase of optical bandwidth followed by a stagnation of spectral broadening, occurs just at the end of the nonlinear fiber. Furthermore, the input pulses should be transform limited, as shorter pulses are expected to have a more favorable ratio of obtained optical bandwidth to side-mode amplification. Under these conditions, an upper limit on the propagation length in terms of L_NL 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 A as a function of initial pulse duration T_FWHM and final RMS-bandwidth Δf_RMS can then be given as:

A < 10 [1 + \frac{6 π^{2}}{ln (2)} {(T_{F W H M} Δ f_{R M S})}^{2}]

The factor of 10 in Eq. (7) takes into account the spectral structure of the side-mode gain. This structure results from the fact, that during spectral broadening, the spectrum develops modulations, that are more pronounced for the principal modes than for the side-modes, see Figs. 2 and 3. The largest peak-to-valley modulation in Fig. 2 is 16 dB, which leads to the factor of 10 in Eq. (7). For the broadened spectra reported in the experimental part of this work, this was about equal to the deepest modulations that have been observed. For spectra with significantly deeper modulations, the prefactor should be adapted accordingly. For example, a conservative estimation would be to replace the factor of 10 by the depth of the deepest dip in the measured spectrum.

Self-steepening, which has been neglected so far, is a higher order nonlinear effect than SPM. The bulk of the spectral broadening and side-mode amplification will therefore be caused by SPM. The split-step method makes it easy to incorporate self-steepening into our numerical model. Its main effect is to make the final pulse shape and spectral envelope asymmetric. We find, that the side-mode amplification through self-steepening is consistently lower than the one caused by SPM at equal spectral broadening, see Fig. 7. So, if we neglect self-steepening, and attribute the spectral broadening fully to SPM as done in Eq. (7), we overestimate the side-mode amplification, and the upper limit represented by Eq. (7) still holds.

Fig. 7 Average side-mode suppression versus spectral bandwidth, with and without self-steepening and self-phase modulation (SPM). All other parameters are the same as in Fig. 2. (a) Comparison of pure self-steepening with pure SPM. (b) Comparison of a mixture of self-steepening and SPM with pure SPM. In all cases, the curve that includes self-steepening is below the one for pure SPM. This means, that for a given amount of spectral broadening, self-steepening amplifies the side-modes less than SPM.

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The effect of Raman gain on side-mode amplification has been studied in [13]. It was found, that concerning side-mode amplification, SPM is the dominant mechanism over the Raman effect. Similarly to what we find for self-steepening it can thus be concluded, that even if the Raman effect is strong enough to contribute substantially to the spectral broadening, attributing the complete broadening to SPM over-estimates the side-mode amplification, and Eq. (7) still holds.

Guided acoustic-wave Brillouin scattering (GAWBS) has been shown to be a very efficient gain mechanism in PCFs, transferring optical power from one spectral mode to another, if their relative optical frequency difference is in resonance with an acoustic mode of the PCF core [26]. The bandwidth of the acoustic resonances, and thus of the optical gain profile, however, is in the range of some MHz only. So, even in the situation, that a side-mode is inside the gain profile of a principal mode, the side-mode amplification can only be driven by one principal mode at a time. For our experimental conditions described in the next section, we estimate, that even under resonant conditions the gain provided for the side-modes is more than seven orders of magnitude lower than the gain provided by SPM, assuming the same gain coefficient as in [26].

As mentioned in Section 1, SHG can be used to convert a filtered frequency comb into the visible spectral range. The second harmonic polarization P²^ω(t) = 2ε₀χ⁽²⁾E²(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(h²) = 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

As shown in Fig. 8, our experimental setup consists of two parts: A filtered frequency comb for astronomical applications (astro-comb) and a heterodyne system used for side-mode characterization. The source of the astro-comb is a mode-locked ytterbium fiber laser with a repetition rate of 250 MHz and a center wavelength of around 1030 nm. The mode spacing of the comb is increased in two steps using concatenated Fabry-Pérot cavities (FPCs) with a finesse of about 400. In order to stabilize the FPCs, a Nd:YAG continuous wave (cw) laser is phase locked without offset to a mode of the source-comb. Then the FPCs are stabilized to the cw laser, using an orthogonal polarization, with the Pound-Drever-Hall scheme [14]. In between the FPCs an ytterbium-doped single-clad fiber amplifier compensates the loss of average power due to the filtering. The filtered frequency comb is amplified to up to 8 W of average power by an ytterbium-doped double-clad fiber amplifier, that also shifts the center of the spectral envelope to about 1060 nm. The optical pulses are compressed to 100 fs FWHM pulse duration by a combined grating and prism compressor, and frequency doubled in a 3 mm long LBO crystal with a conversion efficiency of up to 3%. The frequency doubled light is coupled into a tapered PCF with a 30 cm long uniform taper waist and a core diameter of around 540 nm in the tapered section [15]. The coupling efficiency is about 50%, resulting in up to 80 mW of spectrally broadened light with a −20 dB-bandwidth of 120 to 240 nm. A half-wave plate is used to control the polarization at the input of the PCF.

Fig. 8 Basic experimental setup of which several variations have been used in the experiments described here. Comb: 250 MHz mode-locked ytterbium-doped fiber frequency comb; Nd:YAG: Neodymium-doped continuous-wave laser; ECDL: External-cavity diode laser; Yb-amp: Ytterbium-doped fiber power amplifier; GPC: Combined grating and prism compressor; SHG: Second-harmonic generation stage; HWP: Half-wave plate; PCF: Tapered photonic crystal fiber.

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In the heterodyne system, an external cavity diode laser (ECDL) is used as a tunable cw source, which is locked with a +20 MHz offset to one of the comb modes. For this purpose, a fast analog phase lock loop is employed, enabling a tight lock with a line width of the locked beat signal below 1 Hz [27, 28]. This extremely low line width enables an excellent signal-to-noise ratio in the heterodyne measurements, allowing the detection of even extremely weak side-modes.

The cw laser is amplified to 3 W of power in a two-stage ytterbium-doped fiber amplifier, and then frequency doubled in a single pass through a 5 mm long KNbO₃-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.

This setup was modified in several ways to characterize the effects of the individual nonlinear steps separately, and to adapt the system to the intended applications: First, the PCF and both SHG stages were removed, to measure the side-mode suppression in the IR. The effect of the SHG on the side-mode suppression was then investigated by reinstalling the SHG stages. Next, the PCF was added to see the effect of SPM. Finally, a third FPC was added to further suppress the side-modes. Moreover, different FPC filter ratios were used to characterize the system under the conditions intended for applications.

4. Experimental results

The measured heterodyne signal was analyzed with a radio-frequency spectrum analyzer. We compensated for the slightly frequency dependent gain and the noise level of our detection system, i.e. photodetectors and amplifiers. We observed a small random frequency shift in the range of ±25 Hz on the signal when using the amplifier of the astro-comb at high power, probably caused by insufficient stabilization of the pump diodes. Consequently we limited the resolution bandwidth of our measurements to 91 Hz when working with frequency doubled light. This limited our capability to detect side-modes in the green to a maximum suppression of about 60 to 70 dB, as opposed to 80 to 100 dB in the IR.

We first used the heterodyne system to test the performance of the two FPCs, which were set to an FSR of 2.25 GHz and 18 GHz, respectively. The best choice of filter ratios for FPCs used in series, that gives the lowest calibration line shift, depends on their transverse mode spectrum [12]. In addition, the filter ratios of the individual FPCs cannot be too large in order to provide sufficient seed power for the optical amplifiers in between the FPCs. The measured side-mode suppression in the IR is shown in Fig. 9. Comparing with the computed TEM00 transmission curve of the FPCs, we find very good agreement except for the side-modes in the vicinity of the TEM01/TEM10 spatial modes of the cavities. This is the case at an offset from the principal mode of +750 MHz and −1.5 GHz for FPC1, and most significantly, at +2 GHz with an excursion of 9 dB for FPC2. All other deviations are probably due to the fact, that the finesse of FPC2 is slightly lower than derived from the mirror specifications.

Fig. 9 Mode suppression at 1064 (532) nm with two filter cavities of 2.25 GHz and 18 GHz FSR: Beat measurement before SHG (red circles), theoretically expected values before SHG (hollow circles) and beat measurement after SHG (blue triangles). Inset: Larger section of the comb structure with principal modes and side-modes. The red square shows the modes under investigation. Bottom: Difference of side-mode suppression before and after SHG. The dashed line marks the 6 dB-level.

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Adding the SHG stages shown in the setup of Fig. 8 resulted in a configuration similar to the one that was used for calibration of an astronomical spectrograph in 2009 [29]. To characterize the effect of SHG on side-mode suppression, the heterodyne signals before SHG and after SHG were compared. As shown in Fig. 9, we find the expected increase by 6 dB in power within the estimated accuracy of the heterodyne measurement of ±1 dB. The only exception is the mode at +2 GHz affected by the higher order spatial transmission of FPC2.

For the characterization of SPM, the two FPCs were tuned to 2 GHz and 14 GHz respectively, leading to a measured suppression in the IR of the strongest side-mode of 38 dB, which is 2 dB lower than expected from the mirror specifications. Adding the tapered PCF as shown in Fig. 8 generates a broad spectral envelope. Its width can be adjusted with the pump current of the power amplifier of the astro-comb. At the maximum current of 4 A we obtained an average power of 60 mW coupled through the PCF and measured a spectral width of 122 nm at the −20 dB point. From the corresponding RMS-bandwidth of 23.5 THz we estimate the worst-case side-mode amplification to be 37 dB using Eq. (7) with T_FWHM =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).

In fact this is what was observed with only two FPCs. The principal modes lost a significant amount of power to the side-modes and were depleted. In addition we observed the powers of the side-modes and principal modes after SPM to fluctuate in time, which is probably associated with the spectral structure of the side-mode suppression (see Fig. 3): Fluctuating coupled optical power or rotation of the waveplate results in changing nonlinear phase shifts and therefore in a changing spectral structure of the suppression. This shifts regions with good or bad suppression in and out of the position of the investigated modes. We therefore recorded time traces of the powers of the dominant side-modes at ±2 GHz and ±4 GHz as shown in Fig. 10.

Fig. 10 Heterodyne measurement of the principal (black) and side-mode powers (colored) around 532 nm of the broadened frequency comb. The optical power for SPM was varied by changing the pump current of the Yb-fiber amplifier of the astro-comb (see Fig. 8). The corresponding shift of the center of gravity of the group of lines is shown below each plot. For large currents some of the side-modes are amplified to powers exceeding the principal mode. In this situation we also observe strong polarization dependence of the side-mode amplification as shown at the lower graph. The dashed vertical lines indicate when the half-wave plate in front of the PCF was rotated by 20°. Side-mode color code: +2 GHz, −2 GHz, + 4GHz and −4 GHz. The side-mode suppression before spectral broadening was 32 dB.

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In the application of the broadened frequency comb for spectrograph calibration, such a behavior can be problematic for calibration accuracy. This is because the side-modes are usually too close to the principal modes to be resolved by the spectrograph as individual lines. To estimate the impact on calibration accuracy, we calculated the center of gravity of the measured ensemble of lines. This allows us to determine the expected calibration errors as shown in Fig. 10. These errors are for most applications not acceptable and defy the idea of precise comb calibration.

To improve the side-mode suppression, we added further FPCs to the astro-comb setup. This configuration was used in a recent campaign for the calibration of an astronomical spectrograph [16]. For the test presented here, the first FPC was tuned to 2.25 GHz and two others were operated at 18 GHz. Before SHG the suppression of the two strongest side-modes at ±2.25 GHz was measured to be 78 dB and 76 dB respectively. We assume a 6 dB side-mode amplification caused by the subsequent SHG. After inserting the tapered PCF we could generate a spectral envelope with a −20 dB-bandwidth of 242 nm or 44.2 THz RMS. The side-modes were now observed to be short-term stable and symmetric within ±1 dB with a suppression of significantly better than 50 dB (see Fig. 11). This is in agreement with Eq. (7) with T_FWHM =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].

Fig. 11 Heterodyne beat note measurement at 532 nm of the principal and two strongest side-modes of a frequency doubled and spectrally broadened comb. The signal of the principal mode appears at 40 MHz, while the signal of the side-modes at +2250 MHz (−2250 MHz) appear at 2290 MHz (2210 MHz). The indicated values for the side-mode suppression include corrections for the RF-electronics.

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As the side-mode suppression is expected to have a spectral structure, it is desirable to measure the suppression for several wavelengths across the spectrum. This has been done on an astro-comb system, which has meanwhile been installed at the Vacuum Tower Telescope (VTT) in Tenerife [30]. It uses two FPCs with an FSR of 6 GHz. Heterodyne measurements at 1060 nm / 530 nm of the dominant side-modes at ±250 MHz showed a suppression of 62 dB and 63 dB before SHG. After SHG a suppression of 57 dB was measured on both sides of the principal mode, again confirming the 6 dB amplification by SHG within the estimated accuracy of the measurement of ±1 dB. Under the same conditions the suppression was identical at 1064 nm / 532 nm and at 1054 nm / 527 nm, which supports the assumption of a uniform initial suppression.

Using a tapered PCF, we obtained a −20 dB-bandwidth of 151 nm or 40.6 THz RMS. Applying Eq. (7) with T_FWHM =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.

Fig. 12 Suppression of the strongest side-mode measured at 3 different wavelengths at the Tenerife VTT astro-comb system. The three red squares indicate subsequent measurements without any action taken in between. The black circles are measurements with different polarization directions before the PCF.

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To improve the side-mode suppression on the VTT system, the FPCs have meanwhile been equipped with new mirrors [31]. The cavity finesse was hereby increased to about 3000, leaving the other parameters unchanged. According to the measured values for the finesse presented in [31], we now expect a side-mode suppression of the broadened VTT astro-comb of at least 47 dB. This should permit a calibration accuracy of at least 4.7 kHz.

5. Conclusion

We have presented theoretical models for side-mode amplification in filtered frequency combs through SHG and spectral broadening. The model for SHG predicted an amplification of 6 dB. To describe side-mode amplification through spectral broadening in PCFs, we chose a simplified model that ignores all effects other than SPM. Without dispersion, the RMS spectral bandwidth increases approximately linearly, while the average side-mode suppression decreases quadratically with nonlinear propagation length. Including dispersion, we derived an upper limit on side-mode amplification, which is of particular interest for practical applications, e.g. in astronomy: For spectrograph calibration, the upper limit allows to guarantee a certain calibration precision, regardless of any possible asymmetry of the side-mode suppression around each principal mode. Our model can also be used to describe noise amplification in frequency combs through SPM, which is a related problem as explained in [13], or to describe side-mode amplification through SPM in a fiber amplifier, which has been done in a similar way in [23].

In the experimental part of this work, we presented a technique for measuring the side-mode suppression in the IR and visible spectral range with very high sensitivity based on heterodyning. We characterized the influence of side-mode amplification through SHG and spectral broadening on different astronomical frequency combs. In particular, the 6 dB side-mode amplification through SHG and the estimated worst-case side-mode amplification through spectral broadening could be confirmed.

6. Appendix

The goal of this appendix is to derive Eqs. (4) and (6) from Eq. (3). We are doing this for the limiting case of large side-mode suppression. For this purpose, it is convenient to define B ≡ 1 − h and compute the leading order in B. The filter ratio is set to m = 2, and the carrier frequency to ω_c = 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 R_i = (1 + h)²/(1 − h)² ≈ 4/B². Using Eq. (3), the powers of the n^th principal mode and of the n^th side-mode are given by P_n = S(2nω_r) and S_n = S((2n + 1)ω_r), respectively. To describe the average side-mode suppression R_avg after nonlinear propagation, we average the relative side-mode power S_n/P_n, weighted with the corresponding principal mode power P_n, and invert the result. This yields $R_{avg} = (\sum_{n = - \infty}^{\infty} P_{n}) / (\sum_{n = - \infty}^{\infty} S_{n})$ . The average side-mode amplification A_avg is then given by:

A_{avg} = \frac{4}{B^{2}} \frac{\sum_{n = - \infty}^{\infty} S_{n}}{\sum_{n = - \infty}^{\infty} P_{n}}

Similarly, the RMS-bandwidth Δf_RMS can be expressed as:

2 π Δ f_{R M S} = \sqrt{\frac{\sum_{n = - \infty}^{\infty} {(n ω_{r})}^{2} S (n ω_{r})}{\sum_{n = - \infty}^{\infty} S (n ω_{r})} - {(\frac{\sum_{n = - \infty}^{\infty} n ω_{r} S (n ω_{r})}{\sum_{n = - \infty}^{\infty} S (n ω_{r})})}^{2}}

Taking advantage of the fact that the second term under the square root, which is the spectral center of mass, vanishes, and neglecting the side-modes, Eq. (9) simplifies to:

2 π Δ f_{R M S} = \sqrt{\frac{\sum_{n = - \infty}^{\infty} {(2 n ω_{r})}^{2} P_{n}}{\sum_{n = - \infty}^{\infty} P_{n}}}

A_avg and Δf_RMS are determined by the three sums $\sum_{n = - \infty}^{\infty} S_{n}$ , $\sum_{n = - \infty}^{\infty} P_{n}$ and $\sum_{n = - \infty}^{\infty} {(2 n ω_{r})}^{2} P_{n}$ . To calculate them in a compact way, we use the abbreviations:

f (T) = exp [- (1 + i C) \frac{T^{2}}{2 T_{0}^{2}} + \frac{i}{2} A (T)], A (T) = \frac{2 z}{L_{N L}} exp (- \frac{T^{2}}{T_{0}^{2}})

With the approximation h² ≈ 1 − 2B, and U(z, T) as defined by Eqs. (1) and (2), the power of the n^th principal mode becomes:

\begin{array}{l} P_{n} = S (2 n ω_{r}) & = & \int_{- \infty}^{\infty} [U (z, T) + (1 - B) U ((1 - 2 B) z, T)] e^{+ i 4 π n T / T_{r}} d T \\ \times \int_{- \infty}^{\infty} [U^{*} (z, T^{'}) + (1 - B) U^{*} ((1 - 2 B) z, T^{'})] e^{- i 4 π n T^{'} / T_{r}} d T^{'} \end{array}

\begin{array}{l} = & \int_{- \infty}^{\infty} [f (T) + (1 - B) f (T) e^{- i B A (T)}] e^{+ i 4 π n T / T_{r}} d T \\ \times \int_{- \infty}^{\infty} [f^{*} (T^{'}) + (1 - B) f^{*} (T^{'}) e^{+ i B A (T^{'})}] e^{- i 4 π n T^{'} / T_{r}} d T^{'} . \end{array}

For the sum over all principal modes we obtain:

\begin{array}{r} \sum_{n = - \infty}^{\infty} P_{n} = \sum_{n = - \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} f (T) f^{*} (T^{'}) [1 + (1 - B) (e^{- i B A (T)} + e^{+ i B A (T^{'})}) \\ + (1 - B^{2}) e^{- i B (A (T) - A (T^{'}))}] e^{- i 4 π n (T - T^{'}) / T_{r}} d T d T^{'} \end{array}

\begin{array}{l} \approx \frac{T_{r}}{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} f (T) f^{*} (T^{'}) [1 + (1 - B) (e^{- i B A (T)} + e^{+ i B A (T^{'})}) \\ + (1 - B^{2}) e^{- i B (A (T) - A (T^{'}))}] δ (T - T^{'}) d T d T^{'} \end{array}

where in the last step the sum over n has been replaced by an approximating integral leading to a δ-function. Expanding the term in square brackets yields 4−4B+(1−A(T)²)B² +O (B³), which is independent of B in first order (undepleted principal modes). With the remaining integral over

{| f (T) |}^{2} = exp (- T^{2} / T_{0}^{2})

we readily obtain:

\sum_{n = - \infty}^{\infty} P_{n} = 2 \sqrt{π} T_{r} T_{0}

The sum over the side-modes is treated in the same way with the corresponding square brackets term (1 + A(T)²)B² + O(B³). The remaining integral over $exp (- T^{2} / T_{0}^{2}) B^{2} (1 - A {(T)}^{2})$ is again readily computed:

\sum_{n = - \infty}^{\infty} S_{n} = \frac{\sqrt{π}}{2} T_{r} T_{0} B^{2} [1 + \frac{4}{\sqrt{3}} {(\frac{z}{L_{N L}})}^{2}]

Finally, the sum $\sum_{n = - \infty}^{\infty} {(2 n ω_{r})}^{2} P_{n}$ can be calculated in a similar manner using $\int_{- \infty}^{\infty} p^{2} exp (i p x) d p = - 2 π δ^{″} (x)$ , where δ″(x) denotes the second derivative of the Dirac delta function:

\sum_{n = - \infty}^{\infty} {(2 n ω_{r})}^{2} P_{n} \approx 16 {(\frac{2 π}{T_{r}})}^{2} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} f (T) f^{*} (T^{'}) \sum_{n = - \infty}^{\infty} n^{2} e^{- i 4 π n (T - T^{'}) / T_{r}} d T d T^{'}

\approx - 2 T_{r} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} f (T) f^{*} (T^{'}) δ^{″} (T - T^{'}) d T d T^{'}

= - 2 T_{r} \int_{- \infty}^{\infty} f (T) [\frac{d^{2}}{d T^{2}} f^{*} (T)] d T

= 2 \sqrt{π} \frac{T_{r}}{T_{0}} [\frac{1}{2} + \frac{C^{2}}{2} + \frac{C}{\sqrt{2}} \frac{z}{L_{N L}} + \frac{2}{3 \sqrt{3}} {(\frac{z}{L_{N L}})}^{2}]

Inserting the results for $\sum_{n = - \infty}^{\infty} S_{n}$ , $\sum_{n = - \infty}^{\infty} P_{n}$ and $\sum_{n = - \infty}^{\infty} {(2 n ω_{r})}^{2} P_{n}$ into Eqs. (8) and (10) yields Eqs. (6) and (4).

Acknowledgments

We gratefully acknowledge the support of the European Southern Observatory (ESO), particularly Gaspare Lo Curto, Antonio Manescau and Luca Pasquini. We are equally indebted to the Kiepenheuer Institut für Sonnenphysik, notably to Hans-Peter Doerr, Thomas Kentischer and Wolfgang Schmidt, for letting us characterize their astro-comb system before permanent installation on the VTT site on Tenerife.

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Nonlinear amplification of side-modes in frequency combs

Abstract

1. Introduction

2. Theoretical model

3. Experimental setup

4. Experimental results

5. Conclusion

6. Appendix

Acknowledgments

References and links

Cited By

Figures (12)

Equations (21)

Optics Express