# Optical phase-noise dynamics of
Titanium:sapphire optical frequency combs

Qudsia Quraishi,1,4∗ Scott Diddams,2 and Leo Hollberg3 1 Army Research
Laboratory, Adelphi, MD 20783
2National Institute of Standards and Technology, Boulder, Colorado 80305
3Department of Physics, Stanford University, Stanford, California 94305
4Formerly with Department of Physics, University of Colorado, Boulder, CO
80309
∗Corresponding author<EMAIL_ADDRESS>
###### Abstract

Stabilized optical frequency combs (OFC) can have remarkable levels of
coherence across their broad spectral bandwidth. We study the scaling of the
optical noise across hundreds of nanometers of optical spectra. We measure the
residual phase noise between two OFC’s (having offset frequencies
$f^{(1)}_{0}$ and $f^{(2)}_{0}$) referenced to a common cavity-stabilized
narrow linewidth CW laser. Their relative offset frequency $\Delta f_{0}$ =
$f^{(2)}_{0}$ \- $f^{(1)}_{0}$, which appears across their entire spectra,
provides a convenient measure of the phase noise. By comparing $\Delta f_{0}$
at different spectral regions, we demonstrate that the observed scaling of the
residual phase noise is in very good agreement with the noise predicted from
the standard frequency comb equation.

###### keywords:

ultrafast , phase noise , optical reference , frequency comb equation

The development of stabilized optical frequency combs (OFC) allows for
straightforward schemes for comparisons of optical frequencies which are
separated by 100‘s of terahertz. When OFC’s are referenced to stable atomic
transitions, the OFC provides a bridge to evaluate relative instabilities
based on different atomic frequency standards Diddams et al. (2001); Gerginov
et al. (2006); Ma et al. (2004); Newbury (2011). In such an atomic-optical
‘clockwork’, the fractional instability of the (stabilized) OFC must be
sufficiently low that it does not degrade the measurement of the stability of
the atomic transition. In fact, recent work has shown that the frequency
instability of OFC can achieve levels of 10-19 at averaging times of 500
seconds (when compared against another stabilized OFC Ma et al. (2004)).
Building on previous work with Titanium:sapphire Bartels et al. (2004);
Schibli et al. (2008); Stenger et al. (2002), which showed phase-coherence
across the optical spectrum, here, we focus on the optical phase-noise
dynamics on shorter times scales (100 ns $\leq\tau\leq$ 1 s). We explore
factors that limit the noise floor and demonstrate that the measured scaling
of the phase-noise exhibits the scaling expected from the simple frequency
comb equation.

Figure 1: (a) Schematic of the optical frequency comb (OFC) output of two 1
GHz repetition rate TiS mode-locked lasers. One optical mode of each OFC is
referenced, using a phase-locked loop (PLL), to a cavity-stabilized narrow
linewidth CW source (REF) at 657 nm. (b) Schematic of the setup used to
compare $\Delta f_{0}$ = $f^{(2)}_{0}$ \- $f^{(1)}_{0}$ derived from two
spectral regions, $\nu_{n}$ and $\nu_{q}$. Polarizing beam splitters (PBS) and
a waveplate (W) are used to obtain spatial and polarization overlap. (c)
Photodetected heterodyne beat of $\Delta f_{0}$ from one spectral region.

To determine the optical phase-noise attributable to the OFCs we lock two
independent OFCs to the same cavity-stabilized CW optical frequency reference
and measure the residual phase-noise between the two combs [Fig. 1(a)].
Alternatively, one could use two highly-stabilized optical frequency
references to measure the phase fluctuations of individual comb modes. That
approach would add a source of phase noise not attributable to the combs. Such
a scheme is used for comparisons of optical atomic clocks Rosenband et al.
(2008), spectroscopy Cingoz et al. (2012) and for frequency comparisons
Coddington et al. (2007), in which high levels of long-term coherence have
already been demonstrated. While comparing two free-running frequency combs
Schlatter et al. (2007) give some measure of phase-noise, locking the
frequency combs allows one to robustly quantify the noise across a broad
spectral region and compare with theoretical predictions. The excellent
stability of the comb Ma et al. (2004) makes it an excellent reference tool in
spectral regions from microwave and terahertz to optical domains Quraishi et
al. (2005).

Our apparatus consists of passive two mode-locked titanium:sapphire (TiS) ring
lasers having a pulse repetition rate of 1 GHz. These self-referenced lasers
have been discussed in detail elsewhere Bartels et al. (2004) and employ
piezoelectric actuators for cavity repetition rate stabilization. The offset
frequency of both OFC’s stabilized with an 2f-to-3f technique Ramond et al.
(2002) which requires less than an octave of optical bandwidth, in our case,
from approximately 600 nm to 1150 nm. The $n^{\rm th}$ optical frequency mode
is identified as

$\\!\\!\\!\\!\\!\\!\nu_{n}=nf_{\rm rep}+f_{0}$ (1)

where $f_{\rm rep}$ is the repetition rate and $f_{0}$ is the offset frequency
Udem et al. (2002) and $n$ indexes an optical mode and is an element of the
integers of order $10^{5}$. However, when an OFC is referenced to a stable
optical source with frequency $\nu_{0}$ (at 657nm), we can also identify the
optical mode as $\nu_{n}=\nu_{0}+f^{i}_{\rm beat}$, where the RF heterodyne
beatnote between a mode from OFC1 (OFC2) and the CW reference is denoted by
$f^{(1)}_{\rm beat}$ ($f^{(2)}_{\rm beat}$) or $f^{i}_{\rm beat}$ (where
$i=1,2$). Solving for the repetition rate we have, $f^{i}_{\rm
rep}=(\nu_{0}+f^{i}_{\rm beat}-f^{i}_{0})/n$.

Figure 2: Measured residual phase noise (double sideband) between $\Delta
f_{0}$ from two spectral regions, 800 nm and 621 nm (bandwidth $<$10 nm) as
compared to the spectral region at 657 nm (bandwidth $<$3 nm), the left axis
units are rad/$\sqrt{\rm Hz}$. The prediction is for the residual phase noise
at 800 nm, obtained from Eqn. 3.

When we tune the repetition rates of two OFC’s to be equal, we have a uniform
frequency shift ($\Delta f_{0}$ $=f^{(2)}_{0}-f^{(1)}_{0}$) between all of the
modes of the two combs [Fig. 1(a)]. Here, we are interested in the scaling of
the phase noise away from the optical lock point at 657 nm (that is, from an
imposed fixed point Benkler et al. (2005) of the comb). At the lock point, we
measure $\Delta f_{0}$ within a small bandwidth about 657 nm and then compare
it with $\Delta f_{0}$ measured at another spectral region [Fig. 1(b)]. We
observe excellent signal-to-noise ratios (SNR) on the $\Delta f_{0}$ signal
because 1 nm of optical bandwidth corresponds to $\sim$1000 modes contributing
to the measured beatnote. Figure 1(c) shows the measured RF heterodyne signal
from OFC1 and OFC2 which are spatially, spectrally and temporally combined.

To calculate the residual phase noise, we begin by expressing each optical
mode in terms of the laser’s free parameters, $f_{0}$ and $f_{\rm beat}$, to
obtain: $\nu^{i}_{n}=r_{n}(\nu_{0}+f^{i}_{\rm beat})+(1-r_{n})f^{i}_{0}$
(where $r_{n}=n/n_{\rm lock}$ and $n_{\rm lock}$ indexes the mode nearest the
CW optical reference) Newbury and Swann (2007). The optical phase-noise signal
between OFC1 and OFC2 for the spectral region $n$ may be expressed as
$\delta\nu_{n}=\nu^{(2)}_{n}-\nu^{(1)}_{n}$,

$\delta\nu_{n}=r_{n}(\Delta f_{\rm beat}-\Delta f_{0})+\Delta
f_{0}=r_{n}\Delta f_{\rm rep}+\Delta
f_{0}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$ (2)

where $\Delta f_{\rm rep}=f^{(2)}_{\rm rep}-f^{(1)}_{\rm rep}$ = 0. A similar
equation may be written for the spectral region around the lock point at 657
nm, and this is denoted by $\delta\nu_{q}$. The relative phase noise power
spectral density (PSD) $S_{\Phi}(f)$ away from the lock point is calculated
from the variance of $\delta\nu_{n}-\delta\nu_{q}$, scaled with respect to the
noise bandwidth, giving

$\displaystyle S_{\Phi,n,q}(f)=(r_{n}-r_{q})^{2}[S_{\Phi,f^{(2)}_{\rm
beat}}(f)+S_{\Phi,f^{(2)}_{0}}(f)$ $\displaystyle+S_{\Phi,f^{(1)}_{\rm
beat}}(f)+S_{\Phi,f^{(1)}_{0}}(f)],$ (3)

where $S_{\Phi,f^{(1)}_{\rm beat}}(f)$ and $S_{\Phi,f^{(1)}_{0}}(f)$ denote
the PSD of the electronic locks and where we have neglected all cross terms
[such as $S_{\Phi,f^{i}_{0}}(f)\bigotimes S_{\Phi,f^{j}_{\rm beat}}(f)$].
Phase noise on the relative offset frequency $\Delta f_{0}$ is a measure of
the combs’ residual optical phase noise. The residual phase noise is the
remaining phase noise between the two combs after common mode noise has been
subtracted. Notably, both OFCs are locked to a common CW reference to better
ensure that noise on the beat between the two OFCs (Eqn. 2) yields noise of
one comb with respect to the other and not simply a measure of different CW
references. Regarding coupling between an individual OFC’s two parameters of
$f^{i}_{0}$ and $f^{i}_{beat}$, we note that the two OFCs have independent
pump lasers and phase-locked loops so ideally there should be little
correlation between the two noise sources on each OFCs. Some coupling between
$f^{i}_{0}$ and $f^{i}_{beat}$ is evident and enhanced by amplitude noise on
the pump source of the Ti:S modelocked laser. Phase-locking $f_{0}$ with an
acousto-optic or electro-optic transducer in the pump beam path significantly
reduces the coupling Quraishi (Ph.D. dissertation, University of Colorado,
2007).

Figure 3: In-loop phase noise of the phase-locks of each optical frequency
comb obtained by comparing each beatnote against a synthesizer (left axis
units, rad/$\sqrt{\rm Hz}$). The noise from the offset frequencies
$f^{(1)}_{0}$ and $f^{(2)}_{0}$ as well as the noise of the comb mode locked
to the optical reference are shown (each is filtered by a 1 MHz low pass). The
integrated phase on $f^{(1)}_{\rm beat}$ ($f^{(1)}_{0}$) corresponds to 0.11
fs (0.074 fs) of timing jitter. The integrated phase noise for the phase-locks
of OFC2 for $f^{(2)}_{\rm beat}$ ($f^{(2)}_{0}$) is 0.30 fs (0.15 fs).

We measure the relative optical phase noise across the comb (Eqn. 3) by using
the setup shown in Fig. 1(b) to extract $\Delta f_{0}$ from the spectral
region around 800 nm and compare that against $\Delta f_{0}$ extracted from
near the lock point at 657 nm. We observe that the optical phase noise
increases with increasing frequency range away from the lock point (Fig. 2).
The data in Fig. 2 is integrated from 1 Hz to 1 MHz, where we obtain 1.4
radians for the 800 nm data (the lock point contribution to this value is 0.9
radians). In order to obtain high resolution across all the Fourier
frequencies from 0 Hz to 1 MHz, the phase noise traces shown in this work were
taken by stitching together traces from smaller frequency spans where the
spans initially extended less than 1 kHz and ended at a 1 MHz span.

We can also simply measure the noise of each lasers’ lock to the reference
separately (left hand side of Eqn. 3). Once we measure $f^{i}_{0}$ and
$f^{i}_{beat}$, we can use Eqn. 3 to predict the phase noise of comb modes
($\delta\nu_{n}$) away from the lock point at 657 nm. Using the results shown
in Fig. 3 for the in-loop PSD for all the phase-locks, we plot one typical
predicted phase noise in Fig.2. The predicted phase noise (Fig.2) is in good
agreement with the measured spectral densities across the frequency range. We
find comparable agreement for all of our data sets. Discrepancies arising
above 55 kHz are due to the prevalence of noise outside our PLL bandwidth.

Comparing Fig. 2 and Fig. 3, it is clear that features in the relative noise
traces can be ascribed to features in each of the phase-locks. For instance,
we see that $f^{(1)}_{\rm beat}$ ($f^{(2)}_{0}$, $f^{(2)}_{\rm beat}$)
dominates the noise in the vicinity of 1 kHz (5 kHz, 46 kHz). Equation 3 gives
equal relative weight to contributions from each of the phase-locks. The phase
noise profiles provide useful information about the laser’s phase-noise
dynamics Paschotta et al. (2006). Not surprisingly, tighter phase-locks would
lead to reduced optical phase noise ($\Delta f_{0}$ is a measure of the
optical phase noise). The two lasers having differing noise profiles seen in
Fig. 2 and Fig. 3 because they are pumped with different pump sources, have
different electronic locking bandwidths and are situated in different parts of
the laboratory.

Ideally, the phase noise would be reduced to the level of the shot noise,
which is already the case for frequencies above $\sim$1 MHz. The majority of
the noise we observe is in the frequency range below 1 MHz range. In
Ti:sapphire lasers there is a strong correlation between the pump laser’s
amplitude noise and the optical phase noise, particularly on $f_{0}$ Holman et
al. (2003). Indeed, for the data presented in Fig. 3, the phase noise is
relatively high near 50 kHz due to increased amplitude noise present on the
pump source of both OFC’s Quraishi (Ph.D. dissertation, University of
Colorado, 2007). However, other noise contributions are also important
including: photodetection noise (which can limit our SNR), electrical noise
(ground loops, in-loop electronic noise), vibrational effects (prominent near
30 Hz), servo system gain-bandwidths (60 kHz, limited by the PZT for locking
$f_{\rm beat}$ ) and thermal fluctuations (both within the Ti:S laser
cavities and in their non-common-mode optical paths between the lasers).
Excessive noise on $f_{0}$ causes optical phase noise and hence, can effect
the $f_{\rm beat}$ lock.

To characterize the scaling of the optical phase noise over almost 300 nm, we
integrated the phase noise at discrete points in the spectrum using optical
bandwidths of a few nm and RF bandwidths of 1 MHz. For each of the traces
shown in Fig. 2 and 631 nm, 704 nm, 716 nm and 900 nm, we obtain the
integrated noise values, as shown in Fig. 4. Since one comb mode is locked to
the reference point at 657 nm, the noise at this lock point is measured by
integrating the PLL error signals of $f_{0}$ and $f_{\rm beat}$ . A linear
trend, as predicated by Eqn. 3, is clearly evident [because the integrated
noise $\sim[S_{\Phi,m,q}(f)]^{1/2}$ is proportional to $(r_{n}-r_{q})$]. A
synthesizer was used to mix the lock signals to DC for the phase noise
measurements and its noise level is below the measured noise levels shown
here. The good agreement between the predicted and measured values demonstrate
that the frequency comb equation can predict the integrated phase noise
dynamics across the entire spectral range of the frequency comb. These results
were reproducible even when measurements are separated by several months. Data
for shorter wavelengths ($<$621 nm) are limited by the spectral bandwidth of
our OFC (which extend down to approximately 600 nm). As seen in Fig. 4, the
phase noise at the lock point is 0.9 rad and increases by approximately 0.45
rad at 800 nm (meaning a culmulative phase noise value of approximately 1.35
rad).

The 2f-to-3f technique which measures the offset frequency and locks this RF
frequency, uses two vastly different spectral regions to derive a feedback
signal (namely a $\sim$2 nm spectral region around both 600 nm and 900 nm).
The phase noise from this lock point ($f_{0}$ ) contributes experimentally to
the optical phase noise between comb modes, (see the discussion of Fig. 2).
That is, when we use the frequency comb equation (Eqn. 1) to predict the phase
noise (Eqn. 3), as one moves away from the lock-point, we observe increased
phase-noise which is well matched to the prediction, as shown in Fig. 4. We
see the residual phase coherence is maintained up to $\sim$30 nm away from the
lock point, however we can still predict the scaling of the noise with Eqn. 3
hundreds of nm away. This demonstrates that the frequency comb equation can
also accurately predict the phase-noise. Certainly, to improve phase coherence
one could use high performance phase-locks, with broadband widths and fast
locking actuator feedback, such as acousto-optic modulators Koke et al.
(2010). However, for applications of the frequency comb where two vastly
different spectral lines are used, such as in the case of comparing optical
clocks Rosenband et al. (2008), averaging the measured beat with the comb
produces a highly frequency stable signal Rosenband et al. (2008),
notwithstanding the lack of phase coherence across the comb. Since the scaling
is predicted to be independent of the lock points (that is $f_{0}$ and $f_{\rm
beat}$ ), a future study could investigate the scaling with respect to the
lock point.

Figure 4: Scaling of the integrated (from 1 Hz to 1 MHz) phase-noise away from
+243 nm (-124 THz) to -36 nm (+26 THz) from the optical lock point at 657 nm
(where the lock-point phase noise of 0.9 radians is subtracted from all the
data). The error bars are not visible on the scale shown. The line is a least-
squares fit to the data.

In summary, we have measured the scaling of the phase noise away from the
comb’s optical lock point. We observe that the majority of the measured phase
noise is attributable to technical noise associated with the comb’s pump
source, servo system gain-bandwidths and environmental effects, which are all
most significant below 1 MHz. Yet, by implementing a scheme whereby we measure
the difference in phase noise between two frequency comb’s locked to a common
reference, we are able to measure the optical phase noise dynamics over the
comb bandwidth of almost 300 nm. The results do not show any fundamental limit
to achieving shot noise performance over a broad range of the measured Fourier
frequencies. Finally, we have shown that the standard frequency comb equation
($\nu_{n}=nf_{\rm rep}+f_{0}$) can predict not only the frequency modes of the
comb but the phase noise scaling as well and we are able to measure the
optical phase noise dynamics over the comb bandwidth of almost 300 nm.

footnote: This work is based on experiments performed in the Time and
Frequency Division at the National Institute of Standards and Technology,
Boulder, CO. This work is a contribution of the US government and is not
subject to copyright in the US.

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