Discusiones importantes y cercanas para los sujetos: contiendas por el lugar de lectura

1.399995 1.400000 1.400005 R-1.4 = (1.7 ± 1.0)×10 -7 F r e q u e n cy R a t i o Time (min) Expected ratio R= M/N = 1.4

Figure 2.15: Ratio of the frequency offsets of the 7th and 5th sideband with respect to the pump laser in the microresonator-based frequency comb shown in figure 2.12. The ratioR= 7/5 = 1.4 is directly measured with the setup presented in figure 2.14 at a gate time of 1 second.

sideband. The measured valueR = 7/5 + (1.7_±1.0) _× 10−7 _{is in good agreement with}

the expected value of R= 7/5.

2.5

Conclusion of equidistance measurements

Two methods for measurement of the equidistance of microresonator based frequency combs have been shown in the previous section. The first method is straight forward and determines a value for the equidistance of the comb’s mode spacing by stabilizing the pump laser frequency and measuring the frequency of two other comb modes. In order to circumvent difficulties of simultaneous measurements with two frequency counters, the “ratio counting method” has been introduced. This method relies on direct measurement of the ratio of two frequency differences between comb modes, which should correspond to a rational number in case of an equidistant frequency comb. Finally, we have intro- duced the value (cf. equation 2.10) as a measure of the equidistance of a frequency comb. Equidistance valuesRmeasured with the “ratio counting method” according to the previous section can be converted to using

= f rf M −fNrf M ₋N − f_Nrf ₋f₀rf N = f_Nrf₋f₀rf M ₋N ·R+ f rf 0 −fNrf · M (M ₋N)_·N , (2.13)

56 2. Equidistance of the generated comb modes

withf_Mrf,f_Nrf andf₀rf being the beat note of the Mth sideband,Nth sideband and the pump laser with the reference comb, respectively. Table 2.1 shows a summary of equidistance measurements at different gate times with the corresponding values for . These -values can be used to calculate a weighted mean that represents the stability of microresonator- based optical frequency combs. Using the standard error σ of the values ¯ as the weight,

the weighted mean ¯w is calculated as

¯ w = P /σ2 P 1/σ2 (2.14) and the corresponding error σw of the weighted mean is derived from

σ2_w = P1

1/σ2

. (2.15)

Taking all measurements from table 2.1 into account leads to a value of the comb equidis- tance of

¯

w = (−0.8±1.4) mHz

measured in a microresonator with a diameter of 172µm and a corresponding mode spacing of 386 GHz. The value for ¯w is measured over a bandwidth of 7 free spectral ranges of

the resonator (2.7 THz bandwidth) and the accuracy of the comb’s equidistance relative to the measurement bandwidth is

σw

2.7 THz ≈5.2×10 −16

Normalization to the optical carrier frequency of 193 THz (_≈ 1550 nm) yields a relative accuracy of the mode spacing equidistance in the order of 7.3_×10−18_{. This value describes}

the stability of sideband beat notes with an unknown light source assuming a stabilized pump laser with known frequency is used for the experiment. Note that 9 data points have been removed from the total of 8382 measurements that are used to determine the weighted mean of in table 2.1. These data points were more that 15 standard deviations off the mean value of the respective measurement and are expected to originate from the pump laser being out-of-lock.

2.5 Conclusion of equidistance measurements 57

Gate time (s) Counts ¯_±σ (mHz) StdDev of (Hz) Method

0.03 217 -33 _±556 8.2 ratio 0.1 223 -80 _±181 2.7 ratio 0.3 293 2.4 _±50.1 0.86 ratio 1 3493 -0.91 _± 5.46 0.32 2 counters 1 3499 3.9 _±10.1 0.60 2 counters 1 98 -40.1 _± 27.4 0.27 ratio 1 179 8.0 _±25.5 0.34 ratio 3 173 5.8 _±12.6 0.17 ratio 10 22 -17.9 _± 15.0 0.070 ratio 30 39 1.65 _± 7.41 0.046 ratio 60 72 -1.88 _± 3.00 0.025 ratio 100 18 1.12 _± 5.98 0.024 ratio 100 42 -0.26 _± 2.69 0.017 ratio 300 14 -0.82 _± 2.83 0.011 ratio Weighted Mean ¯w: -0.8 mHz ± 1.4 mHz - -

Table 2.1: Measured mean values of at different gate timesτ (≡0 is expected for a

perfectly equidistant comb). Mean values ¯and standard error σ are given in the third

column. The second column shows the number of frequency counts for each measurement and the standard deviation of the corresponding distribution is given in the fourth column. The last column states the measurement method (either with two counters or direct measurement of the mode spacing ratio). The total measurement time sums up to 6 h 37 min.

CHAPTER 3

Stabilization and control of microresonator

based frequency combs

60 3. Stabilization and control of microresonator based frequency combs

3.1

Motivation

A light source emitting a large number of equidistantly spaced optical frequencies is an interesting tool for a variety of applications in both basic research and applied sciences. Several applications including channel generation for optical telecommunication or mea- surement of relative optical frequencies do not require a fully referenced and stabilized optical frequency comb and can solely rely on the intrinsic stability of the microresonator modes. However, highly accurate frequency measurements require referencing and sta- bilization to a primary frequency standard. This chapter shows that measurement and control of the offset frequency and the mode spacing of microresonator based frequency combs can be achieved by actuating on the power and frequency of the laser source that is used to pump the microcavity. The demonstration of frequency comb control goes along with an analysis of the intrinsic stability of microresonator based frequency combs, which is affected by intensity and frequency noise of the pump laser light in the microresonator as well as temperature and mechanical stability of the experimental setup. Two different schemes for the stabilization of microresonator based frequency combs are presented. In the first case, stabilization of a comb with a mode spacing exceeding 500 GHz is demonstrated by locking two different comb modes to neighboring comb lines of a conventional fiber laser based frequency comb. The second scheme demonstrates direct referencing of the mode spacing to a primary frequency standard by using millimeter size resonators with a mode spacing below 100 GHz, which is amenable to direct detection with a fast photodiode. The second degree of freedom in both schemes can be directly controlled and stabilized via the frequency of the pump laser, which is constituting one of the comb modes.

3.2

Basic principle of microresonator frequency comb