# Coherent photo-thermal noise cancellation in a dual-wavelength optical
cavity for narrow-linewidth laser frequency stabilisation

Fabian Dawel  Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116
Braunschweig, Germany  Institut für Quantenoptik, Leibniz Universität
Hannover, Welfengarten 1, 30167 Hannover, Germany  Corresponding author:
<EMAIL_ADDRESS>Alexander Wilzewski  Physikalisch-Technische
Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany  Institut für
Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover,
Germany  Sofia Herbers  Physikalisch-Technische Bundesanstalt, Bundesallee
100, 38116 Braunschweig, Germany  Lennart Pelzer  Physikalisch-Technische
Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany  Johannes Kramer
Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig,
Germany  Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten
1, 30167 Hannover, Germany  Marek B. Hild  Physikalisch-Technische
Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany  Institut für
Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover,
Germany  Kai Dietze  Physikalisch-Technische Bundesanstalt, Bundesallee 100,
38116 Braunschweig, Germany  Institut für Quantenoptik, Leibniz Universität
Hannover, Welfengarten 1, 30167 Hannover, Germany  Ludwig Krinner
Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig,
Germany  Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten
1, 30167 Hannover, Germany  Nicolas C. H. Spethmann  Physikalisch-Technische
Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany  Piet O. Schmidt
Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig,
Germany  Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten
1, 30167 Hannover, Germany

###### Abstract

Optical resonators are used for the realisation of ultra-stable frequency
lasers. The use of high reflectivity multi-band coatings allows the frequency
locking of several lasers of different wavelengths to a single cavity. While
the noise processes for single wavelength cavities are well known, the
correlation caused by multi-stack coatings has as yet not been analysed
experimentally. In our work, we stabilise the frequency of a $729\,$nm and a
$1069\,$nm laser to one mirror pair and determine the residual-amplitude
modulation (RAM) and photo-thermal noise (PTN). We find correlations in PTN
between the two lasers and observe coherent cancellation of PTN for the
$1069\,$nm coating. We show that the fractional frequency instability of the
$729\,$nm laser is limited by RAM at $1\times 10^{-14}$. The instability of
the $1069\,$nm laser is at $3\times 10^{-15}$ close to the thermal noise limit
of $1.5\times 10^{-15}$.

##  1 Introduction

Optical Fabry-Pérot cavities are an indispensable tool in precision
experiments like gravitational wave detectors [1, 2, 3], opto-mechanical
systems [4, 5, 6], micro-cavities [7, 8], or optical atomic clocks [9].
Nowadays, the best optical cavities used for optical clocks reach a fractional
length instability of $4\times 10^{-17}$ at $1\,$s [10]. This instability is
limited by the thermal noise of the mirror coating. It has been proposed to
reduce coating thermal noise by employing different materials and coating
stacks [11, 12, 13, 14], and this was experimentally demonstrated by
introducing aSi into a highly reflective SiO2/Ta2O5 coating [15].

Apart from laboratory experiments in well-controlled environments, a rising
number of transportable optical clocks [16, 17, 18, 19, 20, 21, 22, 23] also
have clock laser stabilisation cavities with instabilities below $1.6\times
10^{-16}$ at $1\,$s [24]. Additionally, there are cavity designs for space
applications [25, 26, 27, 28] requiring even smaller, lighter and more robust
setups.

A growing number of spectroscopy experiments use multiple atomic species. The
second species is used for sympathetic sideband cooling on a narrow optical
transition [29, 30, 31, 32, 33] or readout via quantum logic [34, 16, 35, 36,
37]. Therefore, the demand for stabilising multiple narrow-linewidth lasers by
means of an optical cavity is increasing. Separate resonators for each
wavelength are expensive and increase the footprint of the experiment, which
is a challenge for various applications, such as transportable setups or space
applications. One solution is to use highly reflective dual wavelength coating
resonators.

Locking multiple lasers to one cavity with one mirror pair has been
demonstrated in the past. Different groups locked multiple lasers for Doppler
cooling and repumping on the same mirror pair, to reduce the experimental
footprint [38, 39] and to compensate for thermal drifts [40]. Multi-wavelength
coatings are also often used in transfer-cavity approaches [41, 42, 43, 44,
45, 46, 47, 48, 49, 50, 51].

In this work we present to our knowledge for the first time the stabilisation
of two lasers to a single narrow-linewidth cavity, using a dual-wavelength
coating for $1069\,$nm and $729\,$nm and analyse their frequency instability.
The photo-thermal effect [52, 53, 54, 55, 56, 57, 24, 58] was measured in
order to estimate the resulting frequency instability caused by optical power
fluctuations on the mirrors. In addition, the residual-amplitude modulation
(RAM)[59, 60, 61, 62] in the setup was characterised. In the present analysis,
we focus on the correlation effects caused by the cavity on both lasers. For
comparison, we performed measurements of the individual lasers with the other
physically blocked. Coherent cancellation effects of the thermal noise between
the different coating stacks were observed and characterised.

##  2 Experimental setup

In this experiment we used a $50\,$mm long, cubic, ultra-low expansion
(ULE${}^{\text{\textregistered}}$) glass spacer [63]. The mirror substrates
with 0.5” diameter were made from ULE${}^{\text{\textregistered}}$ glass. One
mirror is flat while the other has a radius of curvature of $R_{2}=500\,$mm.
The non-reflective side of each mirror is wedged ($0.5\,$°) to prevent
etalons. The quarter-wave stacks of the coating for the $1069\,$nm light are
on the substrate, with the $729\,$nm coating stack on top. The coating order
was chosen such as to minimise the absorption losses in the coatings and
maximise the transmission of the $729\,$nm light. High transmission is
important for (self-)injection-locking a slave diode to the cavity-filtered
light in order to improve spectral purity [64, 65, 66, 67, 68]. The coating
stacks111Made by Layertec were made of SiO2 and Nb2O5. The finesse was
measured via a ring-down measurement. This resulted in a finesse of $67.2\,$k
and $50.1\,$k for the $1069\,$nm and $729\,$nm cavities, respectively. For
this cavity the main contribution to the thermal noise limit is given by the
mirror substrate, where the main limitation for times between $0.01\,$s to
$10\,$s is from Brownian noise. The calculated instability flicker frequency
floor for the $729\,$nm frequency instability was mod$\,\sigma_{y}=1.5\times
10^{-15}$ using the method shown in [69]. The thermal noise limit for the
$1069\,$nm cavity is around $20\,$% lower than the $729\,$nm thermal noise
floor, when only the $1069\,$nm coating is considered. The difference in
coating thickness has only a small impact since it is not the main contributor
to the thermal noise limit. The power spectral density of the substrate’s
thermal noise of the $1069\,$nm cavity is $2$ ($3$) times larger than the
coating thermal noise when the whole (only the $1069\,$nm) coating stack
contributes to the Brownian noise. In the plots presented below, we always
show the fractional frequency fluctuations of the $729\,$nm thermal noise
limit.

The estimated frequency shift from acceleration is below the thermal-noise
limit for typical laboratory environments. This is achieved by mounting the
spacer via four edges of the cube to obtain high mounting symmetry [63]. The
measured sensitivity to accelerations at $10\,$Hz acceleration modulation is:
$\sigma_{x}=2.8(6)\times 10^{-11}/\text{g}$, $\sigma_{y}=6.9(3)\times
10^{-11}/\text{g}$ and $\sigma_{z}=7.7(5)\times 10^{-11}/\text{g}$. In
principle, insensitivities on the order of $10^{-12}/\text{g}$ should be
feasible with this spacer for two of the three directions, as shown in
previous work [63]. For additional vibrational decoupling of the cavity, it
was placed on an active vibration isolation platform222Tablestable, AVI600-M.
With the given sensitivity, we reach a fractional frequency instability
contribution from vibrations of $<4\times 10^{-16}$ at 1s.

To reduce temperature drifts, the resonator was thermally isolated. Two heat
shields provide passive temperature stability and the outer heat shield was
actively regulated by four peltier elements. The mounting and the heat shields
are shown in [26]. The zero crossing of the resonator’s coefficient of thermal
expansion was measured to be at $28.5(1.0)\,$°C. A 1% fluctuation at a
pressure of $2\times 10^{-6}\,$Pa at the ion pump, will cause a frequency
instability of $<1\times 10^{-16}$.

The $1069\,$nm and the $729\,$nm lasers are both extended cavity diode
lasers333Toptica DLpro. To transfer the cavity’s fractional length stability
to the laser’s fractional frequency stability, we used a Pound-Drever-Hall
(PDH) locking scheme [70, 71, 72]. The modulation frequency for the $1069\,$nm
laser was $12.3\,$MHz, applied with a wedged electro-optical modulator
(EOM)444Qubig, PM7-NIR+w. The $729\,$nm laser was modulated at $12.7\,$MHz
with a Brewster-cut EOM555Qubig, PM7-NIR+BC. We used two branches for feedback
via a PID regulator666Toptica, FALC: the fast signal controlled the laser
diode current and the slow signal the external cavity grating angle. The
optical setup is depicted in figure 1. Both laser beams are superposed in
front of the cavity with a dichroic mirror. The light transmitted through the
cavity was separated with a dichroic mirror to stabilise the transmitted
optical power of each laser individually. For each wavelength, the transmitted
light beam was split and two photodiodes were used to separately detect power
fluctuations. One photodiode signal was used for the control circuit and the
second allowed for an out-of-loop measurement.

Figure 1: Schematic drawing of the experimental setup. The red and the blue
lines show the $729\,$nm laser path and $1069\,$nm laser path, respectively.
Black lines represent electronic connections. PLL=Phase-locked loop;
PID=Proportional-Integral-Derivative regulator [73]

For the frequency instability measurement, both laser frequencies were
compared to a stable reference laser via an optical frequency comb. The comb
was locked to a $1542\,$nm laser with a fractional frequency instability of
$4\times 10^{-17}$ at $1\,$s [10]. Applying a transfer beat [74, 75] we
measured the instability of both lasers using a synchronous frequency counter
in lambda counting mode [76]. By transfer locking both lasers to the frequency
comb [75] via a phase-locked loop, we were able to extract frequency and phase
shifts from the error signal.

##  3 Measurement and Results

###  3.1 Photo-thermal effect

Changes in the cavity resonance frequency can be caused by light power
fluctuations. This is known as the photo-thermal effect [52, 53, 55, 56, 57,
24]. The absorption of light increases the temperature in coating and
substrate, causing thermal expansion. This process results in photo-thermal
elastic (PTE) noise. Temperature changes can also affect the refractive index
of the coating, altering the optical-path length and thus the resonance
condition [54]. This process results in photo-thermal refractive (PTR) noise.
In a dual-wavelength setup, both light beams are partially absorbed in the
mirror coating. Thus, power fluctuations of both affect the cavity’s
resonances. We assume that the modulation of laser light power is slower than
the heat exchange between all coating layers [58]. This means that the coating
layer temperature along the beam propagation direction is time independent,
and that the power absorption of both lasers will cause the same optical
length change for equal amounts of dissipated power.

| Nb2O5 | SiO2 | ULE
---|---|---|---
$\eta$ | 0.2 | 0.17 | 0.17
$\alpha$(1/K) | $5.8\times 10^{-6}$ | $2.2\times 10^{-6}$ | $0\pm 3\times 10^{-8}$
$\kappa$ (W/[Km]) | assumed: 1 | 1.38 | 1.38
Y (GPA) | 68 | 72 | 67.6
C (MJ/[m3K]) | 2.71 | 1.64 | 1.7
$\Phi$ | $4.6\times 10^{-4}$ | $2\times 10^{-4}$ | $1.6\times 10^{-5}$
$\beta$ (1/K) | $1.43\times 10^{-5}$ | $8\times 10^{-6}$ |
n ($1069\,$nm) | 2.25 | 1.46 |
n ($729\,$nm) | 2.29 | 1.46 |
Table 1: Material parameters used. Here, $\eta$ is Poisson’s ratio, $\alpha$
is the coefficient of thermal expansion, $\kappa$ is the thermal conductivity,
Y is Young’s modulus, C is the specific heat per volume, $\Phi$ is the
mechanical loss factor, and $\beta$ the thermorefractive index. The parameters
are taken from [24, 77, 78] and references therein.

The theory of photo-thermal noise (PTN) for Bragg mirrors was first described
in [79]. Here, we use the equations given in the Appendix. Based on the
material parameters (see Tab.1), the optical length change X caused by light
power fluctuations can be calculated for a given mirror. The total length
change X is caused by the PTR effect of the mirror coating
$X^{(\mathrm{ct})}_{\mathrm{PTR}}$, the PTE effect of the mirror coating
$X^{\mathrm{(ct)}}_{\mathrm{PTE}}$, and the PTE effect of the mirror substrate
$X^{(\mathrm{sb})}_{\mathrm{PTE}}$. The total length change $X$ is a complex
number whose absolute value $|X|$ can be interpreted as the mirror
displacement and $arg(X)$ is the phase between the absorbed power change and
the length change. Thus, a change of phase by $180\,$° changes the sign of X.
The overall shift is then calculated by adding all three contributions:

$|X|=|X^{(\mathrm{ct})}_{\mathrm{PTR}}+X^{(\mathrm{ct})}_{\mathrm{PTE}}+X^{(\mathrm{sb})}_{\mathrm{PTE}}|.$
(1)

Since the PTE effect and PTR effect may be out of phase, it is possible to
have a coherent cancellation between the three parts. The formulas can be used
to design a coating with coherent cancellation between the PTR and PTE effect
as shown in [80] to reduce PTN for a certain frequency range of the power
noise spectrum.

The cavity’s length sensitivity to power fluctuations can be measured by
modulating the power of the laser light and hence the absorbed power. For the
optical power calibration, we measured the transmitted laser light on a
photodiode. We further expect that the measured power transmission through the
cavity is proportional to the power circulating in the cavity and therefore to
the absorbed power in the coatings. As the ratio between scattered and
absorbed light is unknown, we cannot infer the absorbed power (see below). The
amplitude and the phase of the laser frequency change can be extracted from
the error signal of the transfer lock to the reference laser and were
converted into an optical length change. This measurement was performed for
power modulation frequencies from $10\,$mHz to $100\,$Hz. We investigated two
cases. In the first, each laser frequency was locked to the resonator
resonance frequency while the other laser was physically blocked. For the
second, we locked both lasers to the resonator’s respective resonance
frequencies and one laser was power modulated, and then vice versa.

Figure 2: Evaluation of the $729\,$nm laser frequency PTN sensitivity with
theory fit to the measurement data. The upper figure shows the sensitivity and
the lower one shows the phase relation. The laser modulated in power during
the measurement is indicated by ”mod.”. The ”alone” indicates that the
$1069\,$nm laser was blocked. The shown measurement points are the average of
three different power modulation settings. The contributions to the overall
$729\,$nm PTN curves are shown for the $729\,$nm power modulation case with
fitted material parameters. The difference in measured displacement between
both lasers is due to the normalisation per transmitted power.

Let us first look at the frequency response of the $729\,$nm laser frequency
as a function of its light power modulation. Fig. 2 shows the corresponding
optical length change of the resonator per transmitted power. The $729\,$nm
laser power modulation showed no measurable difference between $1069\,$nm
light blocked or frequency coupled to the resonator. The shown curves are fits
of material parameters to account for effects such as the dependence on
coating layer deposition[81] or the use of substrate [82]. We therefore use
combination of material parameters as free parameters, i.e.
$P_{\text{abs/trans}}\times\alpha_{\text{ct}}$,
$P_{\text{abs/trans}}\times\beta_{\text{ct}}$, $C_{\text{ct}}$,
$\kappa_{\text{ct}}$ and $P_{\text{abs/trans}}\times\alpha_{\text{sb}}$. Here,
$P_{\text{abs/trans}}$ is the unknown scaling factor between transmission and
absorption. As this is a fully correlated parameter, it cannot be determined
by a fit independent from the material parameters. We fitted the data to the
$1069\,$nm PTN sensitivity and $729\,$nm PTN sensitivity simultaneously,
because the material parameters are the same in both cases. The fit parameters
are shown in Tab. 2. The fitted curves of the substrate show that it does not
contribute much to the $729\,$nm PTN sensitivity due to its low coefficient of
thermal expansion (CTE). The expansion of the coating is the dominant
contribution to the $729\,$nm laser frequency change. Most of the $729\,$nm
power is reflected in the first layers, so that the $1069\,$nm stack does not
make a significant contribution to the PTR noise.

| Literature |  | Fit
---|---|---|---
$\alpha_{\mathrm{ct}}$(1/K) | $3.6\times 10^{-6}$ |  $P_{\mathrm{abs}/\mathrm{trans}}\times\alpha_{\mathrm{ct}}$(1/K) | $1.80(10)\times 10^{-5}$
$\alpha_{\mathrm{sb}}$ (1/K) | $\pm 3\times 10^{-8}$ |  $P_{\mathrm{abs}/\mathrm{trans}}\times\alpha_{\mathrm{sb}}$(1/K) | $1.2(4)\times 10^{-7}$
$\kappa_{\mathrm{ct}}$ (W/[Km]) | $1.2$ |  $\kappa_{\mathrm{ct}}$ (W/[Km]) | $1.04(10)$
$C_{\mathrm{ct}}$ (MJ/[m3K]) | $2.1$ |  $C_{\mathrm{ct}}$ (MJ/[m3K]) | $1.37(16)$
$\beta_{\mathrm{ct}}^{729}$ (1/K) | $3.2\times 10^{-6}$ |  $P_{\mathrm{abs}/\mathrm{trans}}\times\beta_{\mathrm{ct}}^{729}$(1/K) | not fitted
$\beta_{\mathrm{ct}}^{1069}$ (1/K) | $1.54\times 10^{-4}$ |  $P_{\mathrm{abs}/\mathrm{trans}}\times\beta_{\mathrm{ct}}^{1069}$(1/K) | $3.02(10)\times 10^{-4}$
Table 2: Summary of expected material values from the literature and the
material values we obtained from the fits in Fig. 2 and 3. The expected values
were calculated using the values in Tab. 1 and the equations in the Appendix.
$\beta_{\mathrm{ct}}^{729}$ was not fitted because the measurement is not
sensitive to the thermorefractive noise of the $729\,$nm coating.
$\beta_{\mathrm{ct}}^{1069}$ and $\beta_{\mathrm{ct}}^{729}$ were calculated
using the thin-film transfer matrix method described in [83, 84, 85].

Next we measured the effect of a power modulation of the $1069\,$nm laser
light on the frequency response of the $729\,$nm laser frequency sensitivity,
which is also depicted in Fig. 2. The length change caused by $1069\,$nm laser
light is higher compared to the previous case because of the higher
absorption. This absorption increase is due to $1069\,$nm light passing
through the $729\,$nm coating layers, which absorb a certain portion of the
light. We assume that the heat transfer in the coating in axial direction is
faster than the laser power modulation, and that the only difference in PTN
sensitivity for the $729\,$nm laser frequency is due to the amount of
absorption and transmission differences between both lasers. As we scale to
transmitted power, we do not account for this constant difference due to
absorption and transmission. Therefore, we multiplied a constant absorption-
to-transmission-ratio factor of $\approx 6.2$ to the theory prediction fit of
the $729\,$nm laser. This factor was calculated by the mean of the ratio of
the $1069\,$nm and $729\,$nm laser power modulation data from Fig 2. By using
this rescaling we can fit all $729\,$nm and $1069\,$nm results with the same
fit parameters. The frequency and phase behaviour due to $1069\,$nm power
modulation is similar to the $729\,$nm power modulation case. So the
absorption of light in lower layers of the coating does not change the PTN for
the $729\,$nm laser.

Figure 3: Evaluation of the $1069\,$nm laser frequency PTN sensitivity with
fit to measurement data. The upper figure shows the optical length change
measured with the $1069\,$nm laser, while the lower figure shows the phase
relation. The contributions to the overall $1069\,$nm displacement curves are
shown for the $729\,$nm power modulation case with fitted material parameters.
The shown measurement points are the average of three different power
modulation settings.

The $1069\,$nm laser frequency sensitivity due to a modulation of optical
laser power is shown in Fig. 3. Comparing Fig. 3 to Fig. 2, one can see that
the $1069\,$nm PTN sensitivity is two orders of magnitude lower than the
$729\,$nm PTN sensitivity. This is the case for power modulation by the
$729\,$nm and $1069\,$nm laser. The PTE and PTR effect are out of phase and
the amplitude of both effects is nearly identical. This leads to a coherent
cancellation. While the refractive index change of the $729\,$nm stack has an
impact on the optical path length of the $1069\,$ nm light, the $1069\,$nm
stack has no impact on the optical path length of the $729\,$nm light because
the $1069\,$nm stack is positioned behind the $729\,$ nm stack. Thus, the PTR
contribution to the $1069\,$nm laser’s frequency (Fig. 3) change is much
larger compared to that of the $729\,$nm laser (Fig. 2) at the same light
power modulation. The use of the transfer matrix approach implies that the
amount of PTE noise is the same in both cases.

Using the fit results together with measurements of the absorption, loss and
transmittance of the cavity, one can derive lower bounds for the material
parameters. The loss of the $729\,$nm cavity is $39(1)\,$ppm and its
transmission $23(1)\,$ppm measured in only one direction. Our measurements
also showed that the $1069\,$nm cavity has a loss of $39(3)\,$ppm and a
transmission of $11(3)\,$ppm. To obtain the lower boundaries of the material
parameters, we assume that all losses for the $1069\,$nm laser light are
absorption losses (no scattering loss) such that
$P^{729}_{\mathrm{abs}/\mathrm{trans}}<0.59$, which gives us
$\alpha_{\mathrm{ct}}^{\mathrm{meas}}>3.06(17)\times 10^{-5}\,$1/K,
$\alpha_{\mathrm{sb}}^{\mathrm{meas}}>2.0(7)\times 10^{-7}\,$1/K, and
$\beta_{\mathrm{ct}}^{1069,\mathrm{meas}}>5.11(17)\times 10^{-5}$. For
$\alpha_{\mathrm{ct}}$, the measured value is one order of magnitude larger
than the literature value (see Tab. 2). This deviation can be contributed to a
low $\alpha_{ct}^{\mathrm{SiO}_{2}}$ value. As discussed in [85, 82] the
measured values between the bulk material or a thin film can differ because of
microstructure, interface, and the underlying substrate. For the thermal
expansion of Nb2O5, the reported literature values for
$\alpha_{ct}^{\mathrm{Nb}_{2}\mathrm{O}_{5}}$ vary from $-2\times
10^{-6}\,$1/K to $5.8\times 10^{-6}\,$1/K [86, 87, 82]. The thermal refractive
index $\beta_{\mathrm{ct}}^{1069}$ is also higher than the literature value
(see Tab. 2) due to the larger CTE. The thermal expansion of the substrate
$\alpha_{sb}$ is higher than expected for ULE material. The mirrors and the
spacer are of the same kind of ULE glass (not the same batch), so the mirrors
are operated close to the thermal zero crossing. The large discrepancy is most
likely due to the low sensitivity of the fit to the substrate’s thermal
expansion. The fitted material parameters indeed changed by no more than
$2\sigma$ when $\alpha_{\mathrm{sb}}^{\mathrm{meas}}$ was set equal to the
upper end of the literature value of $3.0\times 10^{-8}\,$1/K. The heat
capacitance values exhibit a large spread as is known from the literature[77].
Our value differs by at least $3\sigma$ compared to the closest literature
value. For the thermal conductance, our value is in agreement with the
literature value (see Tab. 2).

Figure 4: Evaluation of fractional frequency instability due to the PTN of
each laser. Figure a) [b)] shows the PTN for the $729\,$nm [$1069\,$nm] laser.
Both have a PTN below the thermal noise limit of the cavity. For reference,
the electronic noise of the photodiodes is also shown.

With the PTN sensitivity determined in Fig. 2 and 3, we can estimate the laser
frequency fluctuations due to laser power fluctuations in the resonator. Fig.
4 shows that the frequency instability for the $1069\,$nm and the $729\,$nm
laser is not limited by PTN. For the measurement, both lasers are power
stabilised using photodiodes behind the cavity. Two additional out-of-loop
photodiodes measure the remaining power fluctuations of the light transmitted
through the cavity. The PTN of both cavities, $729\,$nm and $1069\,$nm, is
mainly limited by the power fluctuations of the $1069\,$nm laser, which can be
explained by the larger absorption-to-transmission ratio for the $1069\,$nm
light compared to $729\,$nm light.

###  3.2 Residual amplitude modulation (RAM)

In the PDH lock the laser light needs to be phase modulated with a fixed
frequency. The demodulation of the reflected light of the cavity converts
laser frequency fluctuations into amplitude fluctuations of an electronic
signal. Additional optical amplitude fluctuations at the modulation frequency
are called RAM and result in an offset of the electronic error signal that can
change with time and thus compromise the frequency stability of the laser. One
cause of RAM is birefringence in the EOM [60, 61]. Another cause is that
parasitic etalons affect the light amplitude of the carrier and sidebands
differently [59, 62]. There are stabilisation schemes which can mitigate the
effect of RAM [88, 89, 90, 62, 91, 92, 93].

We investigated the fractional frequency instability added by RAM for two
cases. First, we showed the fractional frequency instability caused by RAM for
each laser by physically blocking the light of the other laser. Second, we
stabilised both lasers to the cavity to observe cross-correlations which might
increase the amount of RAM compared to the first case.

We measured RAM using a method similar to the one described in [24]. For this,
we measured the laser frequency to voltage conversion (frequency discriminant)
by modulating the PDH error signal with a known voltage and measured the
corresponding frequency change at the frequency comb while the laser was
locked to the cavity. This resulted in $\approx
60\,\frac{\text{kHz}}{\text{V}}$ for both lasers. For the measurement of RAM
we tuned the laser away from resonance (around $150\,$MHz). The voltage
fluctuations were measured and multiplied by the frequency discriminant to get
the frequency instability caused by RAM.

Figure 5: Evaluation of the laser instability. Figure a) shows the instability
of the $729\,$nm laser including the effects of RAM and PTN. RAM clearly
limits the frequency instability of the $729\,$nm laser. Figure b) shows the
instability of the $1069\,$nm laser and the effects due to RAM and PTN. All
shown modified Allan deviations for laser frequency instabilities are with
linear drift subtracted. All errors shown in the RAM and lasers frequency
instabilities only represent the statistical error of that measurement. They
do not include larger errors due to day-to-day fluctuations or measurement
device errors.

In Fig. 5 a) [b)] we see the fractional frequency instability of the $729\,$nm
[$1069\,$nm] laser caused by RAM. For the $729\,$ nm laser, the instability
caused by RAM is above the thermal noise limit, therefore limiting the laser’s
fractional frequency instability. Typically, Brewster-cut and wedged EOMs
introduce less RAM compared to other EOM types [94, 95]. RAM voltage
fluctuations of the order of $10\,$ppm (see, e.g., [95]) would decrease
frequency fluctuations to a level of $10^{-15}$. Therefore, the birefringence
of the EOM is unlikely to be the reason for the high RAM. We see indications
for parasitic etalons due to a correlation between error signal and the
scanned laser frequency far off-resonance. This was measured using a method
similar to that in [59]. We found a modulation which could be caused by an
etalon with a length of $5\,$m. This etalon is most likely located between the
acousto-optical modulator and the optical isolator, therefore including the
EOM (see Fig. 1). We do not expect that the RAM differs for the different
cases since both lasers are phase modulated at different frequencies, which is
corroborated by the near identical frequency instability of the $729\,$nm
laser for the two cases. The slight difference in RAM for the two $729\,$nm
laser cases can be most likely be attributed to temperature, pressure and
humidity fluctuations in the lab over the course of the day and the limited
measurement time for this long etalon. The frequency instability of the
$1069\,$nm laser with the wedged EOM is not limited by RAM. The EOM shows RAM
voltage fluctuations on a level of $1.6(0.1)\,$ppm (at $1\,$s). The
instability caused by RAM is below the thermal noise limit for averaging times
$<10\,$s. We observe a measurable difference in noise caused by RAM, whether
the $729\,$nm laser is stabilised to the cavity, or blocked. As for the
$729\,$nm laser, we do not find evidence for an influence of the second laser
on the RAM of the $1069\,$nm laser.

###  3.3 Fractional frequency instability

The investigation of the laser fractional frequency instability is shown in
Fig. 5. Here, the individually and simultaneously locked laser instabilities
are shown. The instabilities for the $729\,$nm laser in both cases are nearly
indistinguishable and they only separate for long averaging times $>10\,$s.
The laser instability appears lower than the level expected by the amount of
RAM measured (for $1-10\,$s). Typically, etalons causing RAM noise vary slowly
over time for example due to changes in air temperature and other laboratory
conditions. Subsequent measurements can therefore yield slightly different RAM
fluctuations.

For the $1069\,$nm laser the the noise is very similar but the frequencies in
the locked and unlocked cases can differ by a small amount. The difference
between the measurements is limited by differential optical path length
changes through optical fibres, or by small RAM fluctuations caused by the
laboratory conditions changing during the experiments, and therefore differ
between measurements.

##  4 Conclusion

We investigated the possibility of achieving narrow-linewidth lasers at very
different optical frequencies by locking them to the same mirror pair. The
laser instability due to RAM is not correlated between the two lasers. In the
case of PTN, we could see a clear correlation between frequency shift and
power modulation. The light absorption of both lasers contributes to the PTN
of each laser. The effect of PTN on the fractional frequency instability of
the $729\,$nm laser is below $2\times 10^{-16}$($1\,$s) and thus smaller than
the thermal noise limit. For the fractional frequency instability of the
$1069\,$nm laser, the PTN sensitivity was even below $1\times
10^{-17}$($1\,$s) due to a coherent cancellation between PTR and PTE. The
overall laser fractional frequency instability is $3\times 10^{-15}$ for the
$1069\,$nm laser, which is around two times larger than the estimated thermal
noise limit. We attributed the discrepancy to non-stabilised path lengths in
the optical setup. For the $729\,$nm frequency lock, the fractional frequency
instability was limited by RAM caused by a parasitic etalon and it only
reached a level of $1.3\times 10^{-14}$($1\,$s). This can be improved by
eliminating the etalon, by finding the reflective surface, or by suppressing
it using additional optical isolators. In general the thermal-noise floor and
the insensitivity to RAM can be further improved by employing longer cavity
spacers, thereby being able to achieve even lower length instability. In
conclusion, we demonstrated that dual-wavelength coatings provide a
performance similar to single-wavelength coatings in narrow-linewidth laser
stabilisation and can even reduce thermal noise components through coherent
cancellation.

## Funding

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy – EXC-2123 QuantumFrontiers –
390837967 and Project-ID 274200144 – SFB 1227, project B03. This joint
research project was financially supported by the State of Lower Saxony,
Hanover, Germany, through Niedersächsisches Vorab. This project also received
funding from the European Metrology Programme for Innovation and Research
(EMPIR) cofinanced by the five participating States and from the European
Union’s Horizon 2020 research and innovation programme (Project No. 20FUN01
TSCAC). This project further received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 101019987).

## Acknowledgements

The authors would like to thank Thomas Legero for the optical contacting of
the cavity mirrors. We would also like to thank Uwe Sterr and Benjamin Kraus
for fruitful discussions.

## Disclosures

The authors declare no conflicts of interest.

## Data Availability Statement

Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request.

## Appendix

The equations for calculating the photo-thermal noise are given here solely
for the sake of completeness and were taken directly from [24]

$\delta
X_{\mathrm{PTE}}^{\mathrm{(sb)}}=-\frac{\alpha_{\mathrm{sb}}[1+\eta_{\mathrm{sb}}]}{\pi\kappa_{\mathrm{sb}}}\frac{f_{\mathrm{T}}^{\mathrm{(sb)}}}{if}\delta
P_{\mathrm{abs}}\int_{0}^{\infty}\zeta\mathrm{exp}\left(-\frac{\zeta^{2}}{2}\right)F\left[1-\frac{\zeta}{\zeta_{sb}}\right]\,\mathrm{d}\zeta$
(2)

$\displaystyle\delta X_{\mathrm{PTE}}^{\mathrm{(ct)}}=$
$\displaystyle-\frac{\alpha_{\mathrm{ct}}}{\pi\kappa_{\mathrm{ct}}}\frac{f_{\mathrm{T}}^{\mathrm{(sb)}}}{if}\delta
P_{\mathrm{abs}}\int_{0}^{\infty}\zeta\mathrm{exp}\left(-\frac{\zeta^{2}}{2}\right)F$
(3) $\displaystyle\times\biggl{\\{}\gamma_{1}\left[\cosh\left(\zeta
d_{\mathrm{ct}}\sqrt{2}/\omega\right)+R\frac{\zeta}{\zeta_{ct}}\sinh\left(\zeta
d_{\mathrm{ct}}\sqrt{2}/\omega\right)\right.$
$\displaystyle\left.-\cosh\left(\zeta_{\mathrm{ct}}d_{\mathrm{ct}}\sqrt{2}/\omega\right)-R\sinh\left(\zeta_{\mathrm{ct}}d_{\mathrm{ct}}\sqrt{2}/\omega\right)\right]$
$\displaystyle-\gamma_{2}\frac{\zeta}{\zeta_{\mathrm{ct}}}\left[R\cosh\left(\zeta
d_{\mathrm{ct}}\sqrt{2}/\omega\right)+\frac{\zeta_{\mathrm{ct}}}{\zeta}\sinh\left(\zeta
d_{\mathrm{ct}}\sqrt{2}/\omega\right)\right.$
$\displaystyle\left.-R\cosh\left(\zeta_{\mathrm{ct}}d_{\mathrm{ct}}\sqrt{2}/\omega\right)-R\sinh\left(\zeta_{\mathrm{ct}}d_{\mathrm{ct}}\sqrt{2}/\omega\right)\right]\biggr{\\}}\mathrm{d}\zeta$

$\displaystyle\delta X_{\mathrm{PTR}}^{\mathrm{(ct)}}=$
$\displaystyle\frac{\lambda_{0}\beta_{\mathrm{ct}}}{\sqrt{2}\pi\kappa_{\mathrm{ct}}\omega}\delta
P_{\mathrm{abs}}\int_{0}^{\infty}\mathrm{exp}\left(-\frac{\zeta^{2}}{2}\right)\frac{\zeta}{\zeta_{\mathrm{ct}}}$
(4)
$\displaystyle\times\frac{\sinh{\left(\zeta_{\mathrm{ct}}d_{\mathrm{ct}}\sqrt{2}/\omega\right)}+R\cosh{\left(\zeta_{\mathrm{ct}}d_{\mathrm{ct}}\sqrt{2}/\omega\right)}}{\cosh{\left(\zeta_{\mathrm{ct}}d_{\mathrm{ct}}\sqrt{2}/\omega\right)}+R\sinh{\left(\zeta_{\mathrm{ct}}d_{\mathrm{ct}}\sqrt{2}/\omega\right)}}\mathrm{d}\zeta$

Where the variables are defined as

$F=\frac{1}{\cosh\left(\zeta_{\mathrm{ct}}d_{\mathrm{ct}}\sqrt{2}/\omega\right)+R\sinh\left(\zeta_{\mathrm{ct}}d_{\mathrm{ct}}\sqrt{2}/\omega\right)}$
(5)

$R=\frac{\kappa_{\mathrm{ct}}\zeta_{\mathrm{ct}}}{\kappa_{\mathrm{sb}}\zeta_{\mathrm{sb}}}$
(6)

$\zeta_{\mathrm{ct}}=\sqrt{i\frac{f}{f_{\mathrm{T}}^{\mathrm{(ct)}}}+\zeta^{2}}$
(7)

$\zeta_{\mathrm{sb}}=\sqrt{i\frac{f}{f_{\mathrm{T}}^{\mathrm{(sb)}}}+\zeta^{2}}$
(8)

$f_{\mathrm{T}}^{\mathrm{(ct)}}=\frac{\kappa_{ct}}{\pi\omega^{2}C_{\mathrm{ct}}}$
(9)

$f_{\mathrm{T}}^{\mathrm{(sb)}}=\frac{\kappa_{sb}}{\pi\omega^{2}C_{\mathrm{sb}}}$
(10)

$\gamma_{1}=\frac{1}{2}\frac{1-\eta_{\mathrm{ct}}}{1-\eta_{\mathrm{ct}}}\left[1+(1-2\eta_{\mathrm{sb}})\frac{1+\eta_{\mathrm{sb}}Y_{\mathrm{ct}}}{1+\eta_{\mathrm{ct}}Y_{\mathrm{sb}}}\right]$
(11)

$\gamma_{2}=\frac{1-\eta_{\mathrm{sb}}Y_{\mathrm{ct}}}{1-\eta_{\mathrm{ct}}Y_{\mathrm{sb}}}$
(12)

To get an average of the thermal expansion, heat capacitance and thermal
conductance of the coating we weight the expansion by the thickness of each
quarter-wave stack. Here we include the number $N_{\mathrm{H}}$
$(N_{\mathrm{L}})$ of high (low) refractive index stacks of the $1069\,$nm and
$729\,$nm stacks.

$\alpha_{\mathrm{ct}}=\sum_{\mathrm{k}=1}^{N_{L}+N_{H}}\alpha_{\mathrm{k}}\frac{d_{\mathrm{k}}}{d_{\mathrm{ct}}}$
(13)

$C_{\mathrm{ct}}=\sum_{\mathrm{k}=1}^{N_{L}+N_{H}}C_{\mathrm{k}}\frac{d_{\mathrm{k}}}{d_{\mathrm{ct}}}$
(14)

$\kappa_{\mathrm{ct}}=\left(\sum_{\mathrm{k}=1}^{N_{L}+N_{H}}\frac{1}{\kappa_{\mathrm{k}}}\frac{d_{\mathrm{k}}}{d_{\mathrm{ct}}}\right)^{-1}$
(15)

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