1. Frequency Combs
Luke Charbonneau
Department of Physics, University of Colorado at Boulder, 2000 Colorado Avenue, Boulder, Colorado 80309-0390, USA
luke.charbonneau@colorado.edu
Abstract: The international optics community has taken a special interest in developing
improvements and applications for frequency combs since the introduction of stabilized, ultrashort
(femtosecond) pulse trains via mode-locked lasers around the turn of the 21st
century. Mode-locked,
pulsed femtosecond lasers are capable of creating well-stabilized frequency combs, and these combs
have both potential and realized applications in the fields of metrology, astronomy, spectroscopy,
and optical time-keeping. The number of applications that precise frequency combs have in research
and industry will likely increase dramatically by the end of the decade as the extensive ongoing
research in both public and private labs continues.
1. Introduction
Around the turn of the 21st
century, stabilized femtosecond laser pulse trains became a reality enabled by developments
in mode-locking techniques [1]. These developments in stabilized femtosecond pulse generation allowed for the
creation of stabilized frequency combs which have large bandwidths (namely, bandwidths which span an “octave”,
the importance of which will be explained in section 3: “Frequency comb theory”) [2]. This increased bandwidth and
stability has made frequency combs increasingly useful for many new applications over the last several years.
Due to the inverse relationship between pulse length in the time domain and bandwidth in the frequency domain,
the key to the generation of wide-broadband frequency combs is the generation of stable, ultrashort pulses. Recent
advances in passive mode-locking techniques have allowed the generation of sub-10 fs pulses in Kerr-Lens mode-
locked (KLM) solid-state Ti:sapphire lasers [3].
2. Mode-locked lasers
A realistic laser is not perfectly monochromatic, but has a range of
longitudinal modes (figure 1) that oscillate within the boundaries of
the resonator (i.e. the laser’s optical cavity). The allowed modes
(with mode-order, q - usually on the order of ~105
or 106
) for a
certain wavelength of light, λ, in a laser cavity of length, L, is given
by equation (1):
𝐿 = 𝑞
𝜆
2
(1)
Furthermore, the frequency separation between these
longitudinal modes (in a laser cavity with an index of refraction, n)
is given by equation (2):
𝛥𝜐 =
𝑐
2𝑛𝐿
(2)
By “locking” the phases of these different longitudinal modes
together (note: active, AM mode-locking does not rely on phase-
locking and will be explained), ultrashort pulses can then be
generated through various techniques.
All mode-locking techniques fall into one of two categories:
active or passive. The theory and operation of both active and
passive mode-locked lasers will be explained in this section.
Figure 1: The first 6 longitudinal modes of a laser cavity
[4]

2. 2.1 Active mode-locking
Active mode-locking techniques involve the use of an optical modulator (figure 2) (e.g. acousto-optic modulator,
electro-optic modulator, etc.) to modulate either the resonator losses/gains of the laser (AM mode-locking) or the
phase of the light within the resonator (FM mode-locking) [5].
The concept behind active, AM mode-locking is fairly straightforward. The optical modulator only allows pulses
of light to pass (i.e. the modulator reduces the cavity losses) during a specific, periodic time, which leads to the creation
of short (picosecond) pulses (figure 3) [5].
The active, FM mode-locking technique uses a phase-modulator, such as a Pockels cell, which periodically
changes the phase of light passing through it [5]. If the laser cavity and modulator parameters are chosen carefully,
this will lead to the constructive interference of in-phase light, leading to the creation of short pulses.
The pulse lengths that can be created by active mode-locking techniques are generally much longer (picoseconds)
than the pulse lengths that can be generated by passive mode-locking techniques (femtoseconds). This is because the
shortness of the pulse duration is limited by the speed of the electric modulators [6]. However, active mode-locking
techniques are still very useful for applications which require synchronization with an electric driving signal [5].
2.2 Passive mode-locking
Passive mode-locking techniques differ from active mode-locking
techniques in their use of a saturable absorber within the laser cavity
instead of an electrically-controlled modulator. Saturable absorbers
are a special class of materials that absorb low intensity light and
become transparent to light which is of a sufficient enough intensity
to saturate the material [7]. When choosing a saturable absorber, the
following properties of a candidate material must be considered (for
reasons which will become clear in this subsection): the wavelength
range, modulation depth, saturable intensity, and recovery time [8].
Through the insertion of an appropriate saturable absorber into
the laser cavity (figure 4), a noise spike with a sufficient enough
intensity (i.e. a “random” spike created by the constructive
interference of many in-phase modes) will start to saturate the
absorber, thus reducing its losses [8]. As the noise spike is further
amplified in subsequent round trips in the cavity, the spike will
saturate the losses until a steady train of pulses is formed after many
iterations of this process [8]. In this steady-state of operation, the
gain can be saturated to a degree which just compensates for the
losses of the circulating pulse, whereas other weaker pulses (i.e.
pulses of weaker intensity) and background light are simply
absorbed by the saturable absorber - keeping the instantaneous losses
larger than the gain in-between the large pulses (figure 5) [6].
Figure 3: Optical power and losses over time in an
actively mode-locked laser [5].
Figure 2: Basic setup of an actively mode-locked laser
cavity [5].
Figure 5: Optical power and losses over time in a
passively mode-locked laser. Note that compared to the
actively mode-locked pulses, these pulses are narrow and
have clipped “wings” [6].
Figure 4: Basic setup of a passively mode-locked laser
cavity [6].

3. The saturable absorbers used in passive mode-locking can be “fast” or “slow”, that is, the absorber material either
has a quick recovery time or a long one. If the saturable absorber used is “fast”, (which is usually the most desirable)
then the edges, or “wings”, of the large pulses will be clipped [6]. This clipping of the pulse’s edges further shortens
the pulse length.
3. Frequency comb theory
The output of a stable, mode-locked laser forms a comb of equally-spaced frequencies (figure 6), with the spacing
between these frequencies, 𝑓𝑟, determined by the repetition rate (i.e. the inverse of the period between pulses) of the
mode-locked laser [9]. Therefore, mathematically, the nth frequency of a comb is described by equation (3):
𝑓( 𝑛) = 𝑓𝑜 + 𝑛𝑓𝑟 (3)
The 𝑓𝑜 term in equation (3) is an offset frequency that arises from intracavity dispersion and nonlinear effects
[11]. This offset is called the carrier-envelope offset frequency (CEO frequency) and is described by equation (4),
where Δφo is the phase change in the carrier-envelope offset per laser resonator round-trip. [11]:
𝑓𝑜 =
𝛥𝜑 𝑜 𝑚𝑜𝑑 2𝜋
2𝜋
𝑓𝑟 (4)
Therefore, if both the repetition and carrier-envelope offset frequencies are known, then all the frequencies of the
comb are known. The repetition frequency, 𝑓𝑟, is relatively easy to lock to a radio-frequency source [9]. On the other
hand, 𝑓𝑜, the carrier-envelope offset frequency, is much more difficult to determine and stabilize [9].
3.1 Carrier-envelope offset frequency measurement and stabilization
The standard procedure for measuring the carrier-envelope offset frequency requires a comb with a bandwidth which
covers a factor of 2 in frequency (i.e. an octave) [9]. If a pulsed laser is unable to generate a wide-enough bandwidth
on its own, then supercontinuum generation in photonic crystal fibers may be used to achieve the requisite bandwidth
[11]. Then, a self-referencing scheme can be employed to measure the carrier-envelope offset frequency. This self-
referencing scheme measures the heterodyne beat frequency between a doubled frequency at the low end of the comb
and the corresponding frequency at the higher end of the comb; the resulting beat frequency is the carrier-envelope
Figure 6: A power spectral density vs optical frequency
plot for an ideal frequency comb [10].

4. offset frequency [11]. This referencing scheme is performed using a f-2f interferometer, which doubles the lower
frequency with a non-linear crystal and then overlaps the corresponding 2f comb tooth beam with the doubled lower
beam (figure 7) [12]. As shown graphically and mathematically in figure 7, this interferometric setup allows the
determination of the carrier-envelope offset frequency by measuring the heterodyne beat frequency between the
doubled frequency beam from the low end of the comb and the corresponding high-frequency beam at the higher end
of the comb.
Once both the repetition and carrier-envelope offset frequency are known, the CEO frequency can be stabilized
via a feedback loop from the f-2f interferometer and the comb can then be used in a wide-range of applications.
3.2 Noise in frequency combs
Like most laser generated devices, frequency combs are subject to many different types of noise. The problem of noise
within frequency combs is rather complicated as there are many potential sources of this noise (e.g. mirror vibrations,
quantum noise, thermal drifts, etc.) [11]. However, the quantum-limited carrier-envelope offset frequency noise is less
apparent in shorter-pulsed lasers – further necessitating their development. Furthermore, the timing jitter (small
fluctuations in the repetition frequency/period between pulses) in the comb-generating, pulsed laser are a concern, as
the repetition frequency is the spacing between the “teeth” of the comb and they ideally have equal spacing [11].
4. Frequency comb applications
Since the development of femtosecond, stable, passively mode-locked pulsed lasers around the turn of the 21st
century,
the resulting frequency combs have been applied to the fields of metrology, astronomy, spectroscopy, and optical
time-keeping.
4.1 The use of frequency combs in optical frequency metrology
The use of wide-bandwidth, stabilized frequency combs in optical frequency metrology (i.e. an optical ruler) is fairly
straightforward. An unknown frequency is superimposed onto a frequency comb of completely known frequencies
and a stabilized carrier-envelope offset frequency. Then, the lowest beat notes are measured with a photodetector and
reveal the difference between a comb “tooth” and the unknown frequency (figure 8) [11].
Figure 7: A simplified f-2f self-referencing scheme [13].

5. 4.2 The use of frequency combs in astronomical spectroscopy
Optical frequency combs can be employed to better stabilize astronomical spectrometers (figure 9) than traditional
methods [14]. However, even with stabilization via an optical frequency comb, (whose accuracy is limited only by
the accuracy of an atomic clock) the sub-pixel accuracy of astronomical spectrometers is still problematic due to issues
with pixel-to-pixel variations, limited dynamic range in the starlight-detecting CCD chips, and time-varying center
frequencies of starlight [14]. However, if these problems are resolved and red shifts of ~20 kHz become possible to
detect, then optical frequency comb-stabilized astronomical spectrometers will be able to verify the 1a supernovae
predictions that the expansion velocity of the Universe is increasing on the order of 1 cm/s per year [14].
Figure 8: Measurement of an unknown frequency (blue)
with the known frequencies of a stabilized comb [11].
Figure 9: Simplified stabilization scheme of an
astronomical spectrometer by an optical frequency comb
[14].

6. 4.3 The use of frequency combs in optical atomic clocks
The precise measurement of time is necessary for a wide range of applications, but perhaps most importantly for the
discovery of fundamental physics and metrology [15]. Optical frequency combs are now being employed to make
accurate spectroscopic and frequency measurements of atomic transitions which are used in modern atomic clocks
[15].
5. Conclusion
The international optics community has taken a special interest in developing improvements and applications for
frequency combs since the introduction of stabilized, ultrashort (femtosecond) pulse trains via mode-locked lasers
around the turn of the century. Passively mode-locked femtosecond lasers are capable of creating well-stabilized
frequency combs, and these combs have both potential and realized applications in various fields of research and
industry. Many new, unique applications for frequency combs will likely be developed in the next decade as the
extensive ongoing research in both public and private labs continues.
References
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