Experimental setup

For determining the absolute frequency of the frequency comb line, exact knowledge of the laser repetition rate f_REP and carrier-envelope-offset frequency f_CEO are needed. Every stabilized frequency comb can be expressed as

f (n) = nf_REP ± f_CEO (110)

where the carrier-offset-frequency f_CEO represents the shift of the comb from 0 Hz. The sign of the f_CEO in the experiment is determined by the observation of the heterodyne beat as the reference frequency is changed. The determination of the sign will be explained in detail later on. Each frequency comb mode can be controlled and stabilized through the variation of f_REP and f_CEO.

For an absolute frequency measurement, a fully stabilized frequency comb of the OPO can be achieved by stabilizing both the repetition rate f_REP of the Ti:sapphire laser and the carrier-envelope-offset (CEO) frequency f_CEOof the signal pulses oscillating in the OPO cavity. In this

experiment a frequency comb at 1560 nm central wavelength was achieved by synchronously pumping a 4-mirror ring type optical parametric oscillator (OPO) with a 333-MHz repetition rate Ti:sapphire pump laser centered at 800 nm wavelength (see Figure 1). The repetition rate directly scales up with the change of the cavity length which can be locked via cavity-length control. Approximately 20 mW of the depleted Ti:sapphire laser’s power was used to detect the repetition rate with a high speed InGaAs photodiode (DSC40S). We used the third harmonic of f_REP since we did not have a bandpass filter (BPF) for higher order harmonics.

Nevertheless the third harmonic enhanced the sensitivity to f_REP fluctuations and therefore the locking quality. After amplification, the acquired third harmonic was phase-compared with a 1 GHz reference frequency in a frequency mixer (an analog phase comparator). This 1-GHz external reference from a frequency synthesizer was phase-locked to an active 10-MHz H-maser. When the feedback loop was activated, the repetition rate of the laser was locked to the 10-MHz reference source. The acquired error signal from the frequency mixer gave an output depending on the phase error. The output was amplified in a high voltage amplifier by employing Laser Quantum’s TL-1000 [45] unit whose output was connected to two PZTs placed inside the Ti:sapphire cavity for f_REP locking. f_REP was stabilized via the combination of these two fast and slow piezoelectric transducers (PZTs). The tuning coefficients were _∆V^∆fREP

P ZT 1=2 Hz V⁻¹ for the fast PZT1 and _∆V^∆fREP

P ZT 2=20 Hz V⁻¹ for the slow PZT.

Figure 1. Layout of the PPKTP based OPO: OC, output coupler; PD, photodiode; PBS, polarizing beam splitter; DG, diffraction grating; APD, avalanche photodiode; PCF, photonic crystal fiber; BS, beam splitter. The frequencies f_REP and f_CEO were stabilized with two separate control loops.

We observed that f_CEO scaled up with the repetition rate. As the repetition rate changed, a change of f_CEO was also noticed on the order of 11 MHzV⁻¹. When the repetition rate is locked, the phase and group velocity mismatch is still in play. Every optical element placed inside the laser cavity will affect the carrier-envelope-offset via the group velocity and phase

velocity mismatch. When the repetition rate of the Ti:sapphire laser is stabilized, the acquired f_CEOmust be locked to achieve fully stabilized frequency comb. This stabilization was achieved with a feedback applied to PZT3 placed inside the OPO cavity.

The f_CEO of the signal pulses was acquired by heterodyning coherent pump supercontinuum light at 529 nm with non-phasematched pump-signal sum-frequency generation (SFG) light from the OPO [46]. A home-made phase-frequency-detector (PFD) was used for the phase stabilization of the f_CEOto a 10-MHz reference frequency (H-maser). As the phase fluctuations between the reference frequency and the f_CEO were obtained, the corresponding output from the PFD was acquired. Our PFD could measured phase fluctuations of up to ±32π. The large range of the PFD makes it easier for the feedback loop to track large phase changes which can be suppressed via a servo-controller (LB1005 servo controller) whose output is connected with PZT3 placed inside the OPO. Once the frequency comb was fully stabilized, the output coupled signal centered at 1560 nm was coupled into a single mode fiber and combined on an InGaAs biased detector (Thorlabs DET01CFC) with a common CW laser wavelength for heterodyne beat detection (see Figure 2).

Figure 2. (a) The OC broadband pulse from OPO in comparison with a CW light, (b) The heterodyne beat between the CW laser and OPO for absolute frequency measurement.

The CW laser wavelength was frequency doubled and locked to a Rb D₂ transition line. From the mode number determination and the beat frequency an absolute frequency of the Rb D₂ transition line could therefore be calculated. The beat between the CW laser locked to a Rb transition and the frequency comb spectrum from the OPO at 1560-nm are shown in Figure 2 (b). The bandwidth of 3 MHz was measured because of the wide Rb transition line used for the CW laser’s wavelength locking and the additional noise contribution because of the feedback loop.

In Figure 3 the scheme for the optical frequency measurement of a frequency doubled 1.6 µm DFB-CW-laser [47] is shown. The CW laser was stabilized to a D₂transition line by absorption spectroscopy. The CW laser mode was heterodyned against one OPO comb line resulting in a heterodyne beat frequency. As mentioned already, if the carrier-envelope-offset, repetition rate

Figure 3. Measurement scheme for the absolute frequency measurement of Rb-transition sta-bilized CW laser. DFB is 1560-nm laser, whose second harmonic frequency locked to the Rb transition line; SHG: second-harmonic generation; FRU: frequency referenced unit; PD: photo-diode.

and the beat frequency with the comb are known, an absolute frequency of the CW laser from the OPO frequency comb mode number can be extracted.

Figure 4. (a), (c), (e) represents f_REP, f_CEO and f_BEAT raw data and (b), (d) and (f) represent fractional instabilities accordingly calculated from Allan variance at different gate times.

To ensure that only common wavelengths were heterodyned on the detector, a bandpass filter

was used. The acquired beat note was increased in strength by optimizing waveleplates, band-pass filters and polarizations. We managed to increase the strength of the beat note between one comb tooth and the Rb locked CW laser so that the four channel Menlo counter FXM50 was able to measure it. The acquired 20-dB signal to noise ratio (RBW=100 kHz) beat note was used to determine the comb number and then to calculate the absolute frequency of the Rb transition. We simultaneously counted the repetition rate f_REP, carrier-envelope-offset f_CEO and the beat note f_BEAT frequencies. The fractional frequency instabilities or Allan variance of the measured frequencies are presented in Figure 4. The left side of Figure 4 shows the raw data of the f_REP, f_CEO and f_BEAT measurements correspondingly and the right side presents the calculated Allan variance of each data set. The fractional frequency instability was calcula-ted for 1-second gate time. From the raw data the Allan variance for other gate times up to 1000 seconds was calculated. The frequency fluctuations were measured simultaneously using a multi-channel digital frequency counter (Menlo frequency counter FXM50). The FXM50 coun-ter is a Π-type councoun-ter which was measuring the frequency wihout averaging it. The data was recorded for 1-s gate time. From the measurement the Allan deviation was extracted for diffe-rent gate times. The difference between the Π-type and Λ-type counters explained in Chapter 4.

The fractional instability of the repetition rate and carrier-envelope-offset frequency were cal-culated by dividing their Allan variance by the repetition rate f_REP harmonic (we detected the 45th harmonic) and the 10-MHz reference frequency for the fREP locking accordingly. The measured frequencies of f_REP and f_BEAT can be stabilized to 10⁻¹² level for a 1-s gate time.

The locking of f_REP was limited by the RF oscillator used as a reference source for all locking loops. The f_BEAT fractional stability was calculated in optical domain (Allan variance divided by 384.228-THz frequency). In the case of the carrier-envelope-offset f_CEO the fractional insta-bility was as low as 10⁻¹⁵. It is more important to achieve better stability of f_REP than the stability of f_CEO. For example, the fractional instability for mode number of 1,000,000 will be much higher since the measured fractional instability of f_REP must be multiplied by the mode number. As a result, comb instability is mainly affected by the quality of f_REP than the f_CEO locking and by the H-Maser’s stability [48].

With the frequency counter we were able to calculate fractional fluctuations of the measured frequencies from the absolute values. These absolute frequencies were used to calculate the mode number of the frequency comb line. From the beat note the optical frequency of the Rb transition can be expressed as

f_Rb

2 = nf_REP ± f_CEO± f_BEAT (111)

where f_Rb = 192.114 THz since the CW laser frequency was locked to the ⁸⁷Rb D₂ F =2-3 transition line. When we heterodyne the modes, the beat can appear on both sides of the comb line tooth. For this reason a sign determination is needed. If the beat note when the repetition

frequency of the Ti:sapphire laser is increasing is also increasing, the Rb locked laser’s frequency is on the lower side of the comb tooth. The f_BEAT has negative sign. If a repetition rate increase reduces the f_BEAT frequency, it means that the sign is positive i.e. the CW laser’s frequency is higher than the comb line frequency. The CW laser frequency is on the higher frequency side of the comb tooth. For the determination of the f_CEO sign, the reference frequency for the f_CEO locking was increased by 1 MHz and the frequency of the f_BEAT beat measured. The increase of the beat frequency gave negative sign while a decrease yielded positive sign for f_CEO. Once all signs were clarified and frequencies identified, the mode number n of the frequency comb line could be calculated using

n =

fRb

2 ± f_CEO± f_BEAT

f_REP (112)

n should be an integer number, which from the experimental data was found to be true to about 0.0001. There are many small uncertainties which limit the accuracy of determining the exact beat frequency, repetition rate or carrier-envelope-offset frequencies and therefore the mode number n. For example, our CW laser, which was locked by saturation absorption to a Rb line, was not stable (see Figure 5(a)). Figure 5(b) illustrates oscillations which correspond to the fluctuations of the CW laser frequency at 780 nm. The Rb locked CW laser was not stable. The low frequency drift of the f_BEAT was evident for every measurement. We think the variation of the CW laser’s frequency was due to etalon fringes in the laser setup which can shift with temperature fluctuations. Up to a 20-kHz f_BEAT variation was measured. This shift couples into the measurement of the beat note and therefore to the uncertainty of the determination of mode number and the calculation of the Rb transition line. These two beats were measured in different labs with different counters. They were roughly adjusted in order to compare quantitatively the two measurements. The goal at this point was only qualitative.

Once the measurement was running, the beat could stay locked for several hours until the f_CEO of the frequency comb degraded below the 35-dB signal-to-noise ratio (at RBW=100 kHz) necessary for our electronics. The loss of the beat strength was due to the thermal drift, and the temperature of the laboratory. The f_CEO frequency locking could be be quickly reacquired by some careful adjustments. Here we present f_REP, f_CEO and f_BEAT measurements obtained at the same time.