Analysis of laser frequency stability using beat-note measurement V1.0 09 2013

The frequency of even the most stable ‘single frequency’ laser is not really constant in , but fluctuates. Frequency stability measurement The laser’s frequency stability is often described in terms of its spectral linewidth. However, while Beat-note measurement this term provides a simple figure for For a detailed analysis of laser frequency stability, comparison of various lasers, it lacks detail the most common measurement techniques are about the laser frequency noise which is often spectral analysis of the laser transmission critical in laser applications. through a sensitive interferometer (filter) or a beat-note measurement with a reference laser. For a detailed representation of laser frequency The beat-note measurement is a heterodyne stability we use either the Allan of the technique where the laser output is mixed with frequency fluctuations or the frequency noise the output of a reference laser and the combined spectrum. These representations are signal is measured using a photo detector. The complimentary and originate respectively from a photo detector is a quadratic detector with a time-domain description and a Fourier-domain limited bandwidth and will only trace the description of the frequency fluctuations. Both difference frequency between the two lasers; the the Allan variance and the frequency noise so-called beat frequency (provided the beat spectrum can be calculated from a beat-note frequency is within the detector bandwidth). 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

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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 reference laser source would be a stabilized electrical spectrum analyzer and recorded by frequency comb, as this can provide very high frequency counter. frequency stability, and a huge wavelength range An experimental setup for measuring and which can give a suitable beat frequency for the recording the beat frequency is illustrated in fig. photo detector. A less ideal reference laser 1. The laser output is combined with the choice is to use a laser similar to the laser under reference laser using a fiber coupler and test, though this can make it difficult to estimate attenuated to a suitable level before entering the the environments influence on the laser stability. photo detector and finally being recorded using a By varying the sample time of the frequency frequency counter. counter, the beat-note setup allows analysis of For a precise measurement of the laser frequency the frequency fluctuation on a micro-second stability, the reference laser should be much scale as well a measurement of the long term more stable than the laser under test. An ideal stability (fig. 2). The beat-note measurement of laser frequency frequency fluctuations we need to use a different fluctuations reveals a clear temporal correlation description than the average frequency and between the measured data. The standard . Such a description can be deviation within a subset of the measurement is obtained using the Allan variance or the smaller than the standard deviation of the total frequency spectrum of the fluctuations. measurement interval. To make a meaningful description of the statistical process of the

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Allan variance of frequency

fluctuations

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 , which is the square root of the deviation, y(t) Allan variance.

If the reference laser in a beat-note

measurement is known to be superior in stability compared to the laser under test, then the where is the optical frequency and 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 time deviation of the lasers become

where is the sample time and the discrete, The above assumption is reasonable if the lasers normalized frequency deviation are identical and the noise sources that contribute to the laser frequency stability can be

considered independent. This is the case for many noise sources such as noise introduced by a

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

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1000000 -30 X15 module E15 module 100000 C15 module -50

10000 -70

1000 -90

100 -110 Frequency Noise (Hz/rt-Hz) Frequency

10 -130 (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

Frequency noise spectrum

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 If the reference laser in a beat-note spectrum has the unit measurement is known to be superior in stability compared to the laser under test, then the

calculated frequency noise spectrum is a measure

of the tested lasers frequency stability. However, if the two lasers is considered to contribute where 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

In Fig. 4 are some examples of the frequency

noise spectrum calculated from beat-note

experiments between identical type 1550 nm fiber laser modules. All the measurements are using a X15 AcoustiK module as the reference where is the sample time, N the number of laser. samples, the n’th sample of the beat-

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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

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

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