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Analysis of Laser Frequency Stability Using Beat-Note Measurement V1.1 09 2016

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Analysis of Laser Frequency Stability Using Beat-Note Measurement V1.1 09 2016 Analysis of laser frequency stability using beat-note measurement V1.1 09 2016 The frequency of even the most stable ‘single frequency’ laser is not really constant in time, Frequency stability but fluctuates. measurement The laser’s frequency stability is often described Beat-note measurement in terms of its spectral linewidth. However, while For a detailed analysis of laser frequency stability, this term provides a simple figure for the most common measurement techniques are comparison of various lasers, it lacks detail spectral analysis of the laser transmission about the laser frequency noise which is often through a sensitive interferometer (filter) or a critical in laser applications. beat-note measurement with a reference laser. The beat-note measurement is a heterodyne For a detailed representation of laser frequency technique where the laser output is mixed with stability we use either the Allan variance of the the output of a reference laser and the combined frequency fluctuations or the frequency noise signal is measured using a photo detector. The spectrum. These representations are photo detector is a quadratic detector with a complimentary and originate respectively from a limited bandwidth and will only trace the time-domain description and a Fourier-domain difference frequency between the two lasers; the description of the frequency fluctuations. Both so-called beat frequency 휈퐵 (provided the beat the Allan variance and the frequency noise frequency is within the detector bandwidth). spectrum can be calculated from a beat-note measurement of the frequency fluctuations. laser under test Agilent 53230A frequency counter variable photo attenuator detector 3 dB coupler reference laser Fig. 1: Beat-note measurement setup NKT Photonics – Blokken 84, 3460 Birkerød Denmark – Phone: +45 4348 3900 – www.NKTPhotonics.com 1/5 2 Sample time: 100 µsec X15 module E15 module C15 module 1 Beat frequency (MHz) frequency Beat 0 0 1 2 3 4 5 6 Time (s) 40 Sample time: 0.1 sec X15 module E15 module 30 C15 module 20 10 Beat frequency (MHz) frequency Beat 0 0 1000 2000 3000 4000 5000 6000 Time (s) Fig. 2: Short term and long term beat-note measurement between a Koheras X15 AcoustiK reference laser module and various other 1550 nm modules The beat frequency can be monitored on an under test, though this can make it difficult to electrical spectrum analyzer and recorded by a estimate the environmental influence on the frequency counter. laser stability. By varying the sample time of the An experimental setup for measuring and frequency counter, the beat-note setup allows recording the beat frequency is illustrated in fig. analysis of the frequency fluctuation on a micro- 1. The laser output is combined with the second scale as well as a measurement of the reference laser using a fiber coupler and long term stability (fig. 2). attenuated to a suitable level before entering the The beat-note measurement of laser frequency photo detector and finally being recorded using a fluctuations reveals a clear temporal correlation frequency counter. between the measured data. The standard For a precise measurement of the laser frequency deviation within a subset of the measurement is stability, the reference laser should be much smaller than the standard deviation of the total more stable than the laser under test. An ideal measurement interval. To make a meaningful reference laser source would be a stabilized description of the statistical process of the frequency comb, as this can provide both a very frequency fluctuations we need to use a different high frequency stability and a huge wavelength description than the average frequency and range, which can give a suitable beat frequency standard deviation. Such a description can be for the photo detector. A less ideal reference obtained using the Allan variance or the laser choice is to use a laser similar to the laser frequency spectrum of the fluctuations. NKT Photonics – Blokken 84, 3460 Birkerød Denmark – Phone: +45 4348 3900 – www.NKTPhotonics.com 2/5 푡푖+휏 Allan variance of frequency 휆0 푦̅ = ∫ Δ휈 (푡) 푑푡 푖 푐휏 퐵 fluctuations 푡푖 For a time domain description of the laser frequency noise we introduce the Allan variance In the literature, most people use the Allan of the instantaneous, normalized frequency deviation 휎푦(휏), which is the square root of the deviation, y(t) Allan variance. ∆휈(푡) If the reference laser in a beat-note 푦(푡) = 휈0 measurement is known to be superior in stability compared to the laser under test, then the where 휈0 is the optical frequency and Δ휈(푡) is the calculated Allan deviation is a measure of the instantaneous frequency deviation. The Allan tested lasers frequency stability. However, if the variation is a so-called two-sample variance two lasers is considered to contribute equally to between consecutive measurements with sample the beat frequency stability then the Allan 2 time 휏, 휎푦 (휏) deviation of the lasers become 2 1 2 1 휎푦 (휏) = 〈(푦̅2 − 푦̅1) 〉 휎푦,푙푎푠푒푟 1(휏) = 휎푦,푙푎푠푒푟 2(휏) = 휎푦,푡표푡(휏) 2 √2 where 휏 is the sample time and the discrete, The above assumption is reasonable if the lasers normalized frequency deviation 푦̅푖 are identical and the noise sources that contribute to the laser frequency stability can be 푡푖+휏 1 considered independent. This is the case for 푦̅ = ∫ 푦(푡) 푑푡 many noise sources such as noise introduced by a 푖 휏 푡푖 pump diode or a temperature control. However, some correlation is expected from environmental The Allan variation of the frequency fluctuations noise contributions like room temperature can be calculated using data from the beat-note variations, vibrations and acoustic noise. measurement by using In fig. 3 are some examples of Allan deviation calculated from beat-note experiments between identical type laser modules. The Y10 laser 1E-7 X15 module E15 module 1E-8 C15 module ) Y10 module ( y 1E-9 1E-10 1E-11 AllanDeviation, 1E-12 1E-13 1E-6 1E-5 1E-4 1E-3 0.01 0.1 1 10 100 1000 10000 Sample Time, (s) Fig. 3: Allan deviation calculated from beat-note measurement between 2 identical 1064 nm Y10 laser modules and between a X15 AcoustiK reference laser module and various other 1550 nm modules NKT Photonics – Blokken 84, 3460 Birkerød Denmark – Phone: +45 4348 3900 – www.NKTPhotonics.com 3/5 1000000 -30 X15 module E15 module 100000 C15 module -50 10000 -70 1000 -90 100 -110 Frequency Noise Frequency (Hz/rt-Hz) 10 -130 Phase Noise (dB[rad/rt-Hz]) 1 -150 0.1 1 10 100 1000 10000 100000 Frequency (Hz) Fig. 4 Frequency noise spectra calculated from beat-note measurement between different fiber laser modules with a X15 AcoustiK module used as reference laser modules are 1064 nm fiber lasers whereas the frequency and win(n) is a window function. The other modules are all 1550 nm fiber lasers. window function could be a Hanning function 1 2휋푛 win (푛) = (1 − cos ( )) Frequency noise spectrum 퐻푎푛푛푖푛푔 2 푁 For a Fourier-domain description of laser Often the frequency noise spectrum is frequency noise we introduce the frequency represented as the square root of the power noise spectrum. density spectrum. The power spectral density of the frequency 2 If the reference laser in a beat-note spectrum 푆휈(푓) has the unit 퐻푧 ⁄퐻푧 measurement is known to be superior in stability compared to the laser under test, then the 2 |ℱ(Δ휈(푡))| calculated frequency noise spectrum is a measure 푆 (푓) = 휈 퐵푊 of the tested lasers frequency stability. However, if the two lasers is considered to contribute where Δ휈(푡) is the laser frequency fluctuations equally to the beat frequency stability then the and BW is the measurement bandwidth. We can Power spectral density of the frequency calculate this using the sampled beat-note fluctuation of the lasers become measurement data 1 푁−1 푆휈,푙푎푠푒푟 1(푓) = 푆휈,푙푎푠푒푟 2(푓) = 푆휈,푡표푡(푓) 1 2휏 2 푆 (푘 × ) = |∑(Δ휈 (푛) 휈 푁휏 푁 B 푛=0 In Fig. 4 are some examples of the frequency 2 noise spectrum calculated from beat-note −푖2휋푛푘⁄푁 × win(푛))e | experiments between identical type 1550 nm fiber laser modules. All the measurements are using a X15 AcoustiK module as the reference where 휏 is the sample time, N the number of laser. 휈 (푛) samples, Δ 퐵 the n’th sample of the beat- NKT Photonics – Blokken 84, 3460 Birkerød Denmark – Phone: +45 4348 3900 – www.NKTPhotonics.com 4/5 Relation between Allan variance and frequency noise spectrum Both the Allan deviation and the frequency spectrum of the laser frequency fluctuation are useful methods for describing the laser frequency stability. It is possible to calculate the Allan variance from the frequency noise spectrum using the following relation ∞ sin4(휋휏푓) 휎2(휏) = 2 ∫ 푆 (푓) 푑푓 푦 푦 (휋휏푓)2 0 However, the reverse calculation of the frequency noise spectrum is not generally possible. References / suggested reading [1] Fritz Riehle, “Frequency Standards, Basics and Applications”, Wiley 2004, chapter 3 NKT Photonics – Blokken 84, 3460 Birkerød Denmark – Phone: +45 4348 3900 – www.NKTPhotonics.com 5/5 .
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