# Absolute frequency comb mode number determination in precision by ert634

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```									Absolute frequency comb mode number determination
in precision optical frequency measurements

J. Zhang, Z. H. Lu, Y. H. Wang, T. Liu,
A. Stejskal, Y. N. Zhao, and L. J. Wang
Institute of Optics, Information and Photonics
Max-Planck Research Group and University Erlangen-Nuremburg
91058 Erlangen, Germany

R. Dumke
School of Physical and Mathematical Sciences, Nanyang Technological University, 1
Nanyang Walk Block 5, Level 3, Singapore 637616, Singapore.

We demonstrate a method to determine the absolute mode number of a
frequency comb when it is used to measure accurate frequency of lasers, with-
out the help of wavemeters. Our technique involves changing the repetition
rate of the frequency comb in a two-steps process. Guidelines for choosing
the correct repetition rates for diﬀerent laser linewidths are oﬀered. As a
demonstration, two lasers with diﬀerent linewidths are measured with our
method. The results agree with our measurements with the help of a high
resolution wavemeter. c 2007 Optical Society of America

OCIS codes: 000.2170, 120.3940.

The frequency comb technique is a very useful method to measure the frequency of lasers
accurately.1–6 A frequency comb can be fully characterized by its repetition rate fr and oﬀset
frequency f0 . Both fr and f0 are locked to a stable frequency standard, e.g., a Cs-standard,
such that their values are accurately known. To measure the frequency of an unknown laser
source, the laser frequency fl , is beat with the Nth mode of the frequency comb to generate
a beat frequency fb ,
fl = Nfr ± f0 ± fb .                                (1)
In order to determine fl from Eq. (1), fb needs to be measured, and the mode number N
needs to be determined. Normally a wavemeter with a resolution better than half of the
comb repetition rate will be employed to help determine N. High resolution wavemeters are

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not only expensive, but also need constant calibration to function properly. L. -S. Ma et al.
noted that N can be determined by simply varying the repetition rate of the frequency comb,
without the help of a wavemeter.7 While their method is useful to determine N of a high
repetition rate frequency comb, the method becomes less feasible when the laser frequency
is measured with a low repetition rate frequency comb.
Here, we propose a new method that is simpler and can be used for both high and low
repetition rate frequency combs. Our method is a two-step process. First, a rough estimate of
the mode number is determined by changing the repetition rate of the frequency comb by a
small amount. With this knowledge, the repetition rate of the frequency comb is changed one
more time to calculate the exact mode number. Guidelines for choosing the correct repetition
rates for diﬀerent laser linewidths are oﬀered.
In order to compare with the method of Ref. 7, we summarize their main points here. They
use their frequency comb to obtain two beat frequencies fb1 and fb2 corresponding to two
mode numbers N and N + m. Here m is a small integer that can be controlled and known
by smoothly changing the repetition rate while monitoring the beat note on the screen of an
RF spectrum analyzer. N can then be expressed as

mfr2 ± f02 − (±f01 ) ± fb2 − (±fb1 )
N=                                         .                     (2)
fr1 − fr2
This gives an experimental value of the integer N, denoted Nexp . The diﬀerence between
Nexp and real N, δ = Nexp − N, is obtained by

±fb2 − (±fb1 )
δ=                  .                                (3)
fr1 − fr2
The uncertainties of fb1 and fb2 are determined by the laser ﬂuctuation Δν for the duration of
the measurement. In order to determine N uniquely, it is necessary that δ ≈ 0.1. Hence, the
change in the repetition rate of the frequency comb need to be around 10Δν. For example,
for a laser with Δν = 1 kHz, the repetition rate needs to be changed by 10 kHz. If the laser
wavelength is around 532 nm, then for a frequency comb with repetition rate of 750 MHz,
the mode number N is around 7 × 105 , and the required mode number change m is around
10. This number is easy to be counted on the screen of the RF spectrum analyzer. On the
other hand, with a lower repetition rate frequency comb (fr ≈ 200 MHz), the mode number
N is around 3 × 106 , and the required mode number change m is around 150, a number that
is diﬃcult and cumbersome to count on an RF spectrum analyzer screen.
To start, the signs before f0 and fb are determined beforehand. The sign of the fb can be
determined by observing the relative directional change of the fb with a few Hz change of
fr . With this sign determined, we can determine the sign of the f0 by observing the relative
directional change of fb with small change of f0 . This way all signs can be determined

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unambiguously in Eq. (1). For simplicity, we assume fl = Nfr + f0 + fb in this paper. We
also lock the f0 to a ﬁxed value in the experiments so that we can disregard the contribution
from f0 .
In our proposed method, we ﬁrst obtain a rough estimate of N by changing the repetition
rate fr a small amount such that no mode number change occurs. In this case,

fl = Nfr1 + f0 + fb1 ,
fl = Nfr2 + f0 + fb2 .                                  (4)

Then N can be estimated as
fb2 − fb1
Nest =             .                                 (5)
fr1 − fr2
In order to estimate N as accurately as possible, we need the repetition rate change, Δf12 =
fr1 − fr2 , to be as large as possible without causing a mode number change. For a low
repetition rate frequency comb (fr ≈ 200 MHz), all visible lasers have a mode number N
around 106 , so the maximum repetition rate change allowed is around 100 Hz. To be on
the safe side, we can choose one half of this value, 50 Hz, as our Δf12 . From Eq. (5), the
√
uncertainty of N is δN = 2δfb /Δf12 , where δfb is in the order of the laser linewidth. Thus,
if the laser linewidth is 1 kHz, then δN is approximately 30.
The second step is to improve the accuracy of N. This can be done by changing the
repetition rate of the frequency comb by an even larger amount such that the mode number
changes. Thus,

fl = Nfr1 + f0 + fb1 ,
fl = (N + m)fr3 + f0 + fb3 ,                               (6)

where fb3 is the new beat frequency. The mode number diﬀerence m is an integer and can
be written as,
NΔf13 + (fb1 − fb3 )
m=                        .                            (7)
fr3
Here, Δf13 = fr1 − fr3 is the repetition rate change of the frequency comb in the second
step. In order to calculate m, we substitute N with Nest , and ﬁnd that the main uncertainty
in m is
δm = δNΔf13 /fr3 .                                  (8)
Ideally, δm shall be around 0.1, which sets an upper limit of Δf13 ≈ 0.7 MHz. The lower limit
of Δf13 , according to Eq. (3), is 10Δν = 10 kHz. Finally the mode number N is calculated
to be
mfr3 + (fb3 − fb1 )
N=                       .                            (9)
Δf13
With the knowledge of N, the laser frequency fl from Eq. (1) can be calculated.

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The above procedure is robust for a variety of lasers with Δν ≤ 10 kHz, if we require the
mode number does not change in the ﬁrst step of changing repetition rate Δf12 . For larger
laser ﬂuctuations, a larger Δf12 is needed, which means the mode number will change. In
this case, we can monitor the screen of the RF spectrum analyzer to count a small number
of mode change m . The estimated N is then given by

m fr2 + fb2 − fb1
Nest =                     .                           (10)
fr1 − fr2
All the remaining calculations are the same. For example, by counting one more mode,
m = 1, the measurable laser ﬂuctuation will increase to 20 kHz.
To demonstrate the feasibility of our proposed method, we measured frequencies of two
lasers with diﬀerent linewidths. Our frequency comb (MenloSystems FC-8004) has a repe-
tition rate of around 200 MHz, and an oﬀset frequency 20 MHz.8 Both repetition rate and
oﬀset frequency are locked to a Cs frequency standard (Agilent 5071A). The comb line has
a resolution limited linewidth of 1 Hz, measured with a RF spectrum analyzer with sweep
time of 3 s. The ﬁrst laser is a monolithic isolated end-pumped ring Nd:YAG laser (MISER).
This Nd:YAG laser has an intrinsically high frequency stability and low amplitude noise.
The laser is further locked to a mechanically and thermally isolated high ﬁnesse Zerodur
reference cavity. The laser linewidth is typically less than 10 Hz.9 The laser light is sent
through a 60 m long polarization-maintaining ﬁber to beat with the frequency comb. The
beat note standard uncertainty is measured to be 1.3 kHz at 60 s, which is contributed by
the ﬁber noise. The uncertainty of the beat note can be further reduced by averaging the
measured results over longer period. In order to determine the laser frequency quickly, we
choose an averaging time of 60 s as a good compromise. The MISER has a linear drift rate
of approximately 1 Hz/s, which has a negligible contribution to the beat note uncertainty
with our measurement time. Our measured results are

fr1 = 201, 062.036 kHz, fb1 = 28, 351.080 kHz,
fr2 = 201, 062.031 kHz, fb2 = 36, 231.614 kHz,
fr3 = 201, 051.956 kHz, fb3 = 30, 159.784 kHz,

and are shown in Fig. 1. The average beat frequencies are calculated with 60 s time averaging.
The calculated results are Nest = 1576106.8, m = 79.007, and N = 1575884.249, from
which we take the integer part 1575884. The laser frequency is determined to be fl =
Nfr1 − f0 + fb1 = 316, 850, 453, 890.9 kHz, where the sign before the oﬀset frequency is
determined to be minus for this particular example.
The second laser measured is a 922 nm diode laser that is locked to an ultra stable, high
ﬁnesse cavity placed in another temperature stabilized, vibration isolated vacuum chamber.

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The beat note standard uncertainty is measured to be 9.6 kHz at 60 s. This uncertainty
borders the zero mode change regime, hence we purposely change the repetition rate by
Δf12 = 124 Hz such that m = 1. Our measured results are

fr1 = 201, 060.017 kHz, fb1 = 30, 436.281 kHz,
fr2 = 201, 059.893 kHz, fb2 = 29, 774.308 kHz,
fr3 = 200, 960.034 kHz, fb3 = 30, 719.173 kHz,

and are shown in Fig. 2. The average beat frequencies are again taken with 60 s time
averaging. Since we have one mode number change, Nest = (fb2 − fb1 + fr2 )/(fr1 − fr2 ) =
1616112.258, m = 804.058, and N = 1615996.222, from which we take the integer part
1615996. Finally fl = Nfr1 + f0 + fb1 = 324, 912, 233, 668 kHz.
In conclusion, we propose and demonstrate a new method of absolute frequency measure-
ment using only a frequency comb. We show a two-step procedure for determining the mode
number of the frequency comb that is closest to the frequency of the laser under measure-
ment. Both steps involve changing the repetition rates of the frequency comb. A guideline
for choosing the repetition rates is given. Using this method, we successfully measured two
diﬀerent laser frequencies with diﬀerent linewidths. The measurement results are conﬁrmed
by repeating the measurement with the frequency comb and a high resolution wavemeter.
Our method is particularly useful for a low repetition rate frequency comb, and when a high
resolution wavemeter is not available.
We thank J. Mondia for useful discussions.

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References

a
1. T. Udem, J. Reichert, R. Holzwarth, and T. W. H¨nsch, “Absolute Optical Frequency
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a
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List of Figures

Fig. 1 Counted beat notes between the Nd:YAG laser and the frequency comb under

diﬀerent repetition rates (a) fr1 = 201, 062.036 kHz; (b) fr2 = 201, 062.031 kHz; (c)

fr3 = 201, 051.956 kHz. All measurements were counted with a frequency counter at

1 s gate time. zhangf1.eps

Fig. 2 Counted beat notes between the 922 nm diode laser and the frequency comb

under diﬀerent repetition rates (a) fr1 = 201, 060.017 kHz; (b) fr2 = 201, 059.893

kHz; (c) fr3 = 200, 960.034 kHz. All measurements were counted with a frequency

counter at 1 s gate time. zhangf2.eps

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Fig. 1. Counted beat notes between the Nd:YAG laser and the frequency comb under diﬀerent

repetition rates (a) fr1 = 201, 062.036 kHz; (b) fr2 = 201, 062.031 kHz; (c) fr3 = 201, 051.956

kHz. All measurements were counted with a frequency counter at 1 s gate time. zhangf1.eps

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Fig. 2. Counted beat notes between the 922 nm diode laser and the frequency comb under

diﬀerent repetition rates (a) fr1 = 201, 060.017 kHz; (b) fr2 = 201, 059.893 kHz; (c) fr3 =

200, 960.034 kHz. All measurements were counted with a frequency counter at 1 s gate time.

zhangf2.eps

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