# Widely-tunable, doubly-resonant Raman scattering on diamond in an open
microcavity

Sigurd Flågan  Department of Physics, University of Basel, Klingelbergstrasse
82, Basel CH-4056, Switzerland  Patrick Maletinsky  Department of Physics,
University of Basel, Klingelbergstrasse 82, Basel CH-4056, Switzerland
Richard J. Warburton  Department of Physics, University of Basel,
Klingelbergstrasse 82, Basel CH-4056, Switzerland  Daniel Riedel  Department
of Physics, University of Basel, Klingelbergstrasse 82, Basel CH-4056,
Switzerland  Currently at E. L. Ginzton Laboratory, Stanford University,
Stanford, CA 94305, USA  Corresponding author<EMAIL_ADDRESS>
###### Abstract

Raman lasers based on bulk diamond are a valuable resource for generating
coherent light in wavelength regimes where no common laser diodes are
available. Nevertheless, the widespread use of such lasers is limited by their
high threshold power requirements on the order of several Watts. Using on-chip
microresonators, a significant reduction of the lasing threshold by more than
two orders of magnitude has been shown. However, these resonators lack a
continuous tuning mechanism and, mainly due to fabrication limitations, their
implementation in the visible remains elusive. Here, we propose a platform for
a diamond Raman laser in the visible. The device is based on a miniaturized,
open-access Fabry-Perot cavity. Our microcavity provides widely-tunable
doubly-resonant enhancement of Raman scattering from high quality single-
crystalline diamond. We demonstrate a $>$THz continuous tuning range of
doubly-resonant Raman scattering, a range limited only by the reflective
stopband of the mirrors. Based on the experimentally determined quality
factors exceeding 300 000, our theoretical analysis suggests that, with
realistic improvements, a sub-mW threshold is readily within reach. Our
findings pave the way to the creation of a universal low-power frequency
shifter, a potentially valuable addition to the nonlinear optics toolbox.

##  I Introduction

The development of new laser architectures to create coherent radiation at
arbitrary wavelengths is a continuous endeavor. Solid-state based lasers have
emerged as the leading platform due to their compact design, reliability,
highly stable output and beam quality [1]. Several wavelength regimes,
however, are not directly accessible with semiconductor lasers and require
wavelength conversion in a nonlinear medium.

A promising technique relies on Raman scattering where, upon the creation of a
phonon, photons are red-shifted by the fixed phonon energy. This energy shift
and the efficiency of the process, quantified by the Raman gain parameter,
depend on the material and in particular on the phonons involved [2]. The main
advantage of this approach is that in principle any laser wavelength can be
achieved if a suitable pump laser is available [3]. This gain mechanism
differs from stimulated emission from excited dopants which exhibit a fixed
gain frequency bandwidth [4].

Diamond is particularly well suited to the creation of a Raman-based laser due
to its exceptional properties [5], notably a large Raman gain ($\sim
75\,\textrm{GW}\cdot\textrm{cm}^{-1}$ at 532 nm) [5] and a large Raman shift
($\sim$1 332 cm-1) [6]. This large Raman shift enables wavelengths to be
accessed for which no ideal solution exists in terms of cost, convenience and
output power. Other advantages of diamond in this context are the wide bandgap
of diamond, which prevents free carrier absorption minimizing optical losses
in the visible and the ultraviolet wavelength regimes, and diamond’s high
thermal conductivity, which facilitates efficient heat management.

Diamond Raman lasers have enabled the creation of coherent radiation in exotic
wavelength regimes, e.g. the yellow band [7, 8, 9, 10, 11, 12], where no
common laser diodes are available. High-power diamond Raman lasers have been
implemented across a large range of wavelengths, from the ultraviolet [13]
across the visible [14, 9, 12] and infrared [15, 16, 17, 18, 19, 20, 21, 22,
23], all the way to the mid-infrared [18]. However, current implementations
are limited by their high threshold pump power requirements, typically several
Watts.

Micro- and nanophotonic engineering offers the potential of reducing the
threshold for lasing. Through resonant recirculation of the pump beam in a
cavity with a small mode cross-section, the intensity-dependent Raman gain is
significantly enhanced. In addition, simultaneous coupling of the Raman field
to a second cavity mode boosts the efficiency of stimulated emission. Based on
this doubly-resonant configuration, low-threshold Raman lasers in the infrared
wavelength regime were demonstrated in silica microspheres [24]. Chip-
integration has been implemented using silica microtoroids [25], silicon
waveguides [26, 27] and racetrack resonators [28]. Combining high quality
($Q$) factors with small mode volumes ($V$), silicon photonic crystals have
enabled ultra-low threshold Raman lasing in the nanowatts regime [29].
Recently, molecules adsorbed to silica microtoroids emerged as promising gain
medium [30]. Advances in diamond nanofabrication enabled the demonstration of
integrated diamond Raman lasers using ring resonators at infrared [31] and
near-visible [32] wavelengths. Largely on account of material and
nanofabrication constraints, Raman microlasers at visible frequencies,
however, remain elusive.

Figure 1: (a) Schematic of the plano-concave Fabry-Perot microcavity. The
cavity is formed by two fused silica (SiO2) substrates coated with a
distributed Bragg reflector (DBR). One of the substrates exhibits spherical
microindentations resulting in a Gaussian resonator mode with a small beam
waist ($\sim 1\,\upmu\textrm{m}$). Piezoelectric nanopositioners allow for
spatial and spectral tunability of the cavity mode. The wedged diamond
membrane enables the diamond thickness within the cavity mode to be changed by
lateral positioning. The thickness gradient of the diamond is exaggerated for
enhanced visibility. (b) One-dimensional transfer-matrix calculation of the
cavity mode-structure for a diamond thickness of
$t_{\textrm{d}}=756\,\textrm{nm}$. The presence of the diamond membrane leads
to a nonlinear mode dispersion. At a mirror separation of $\sim
4.18\,\upmu\textrm{m}$, the frequency spacing between modes
$q_{\textrm{eff}}=17$ and $q_{\textrm{eff}}=19$ equals the frequency of the
optical phonon in diamond ($\sim 40\,$THz). (c) In red: measurement of the
reflectivity of the planar DBR revealing a stopband centered at 625 nm with a
bandwidth of $\sim 100$ THz. In blue: experimental cavity spectrum for a fixed
cavity length under $532\,\textrm{nm}$ illumination. The diamond thickness and
mirror separation can be inferred by comparing the spectrum with the simulated
mode structure in (b). Inset: Raman process, depicted as a three-level system.
A photon is converted to a red-shifted photon and an optical phonon of fixed
frequency.

We propose to resolve this conundrum by using a different platform for the
implementation of a low-threshold diamond Raman laser, namely a highly
miniaturized Fabry-Perot cavity [33]. Such microcavities, sometimes referred
to as open microcavities, offer high $Q$/$V$ ratios and thereby promote strong
light-matter interactions [34, 35]. We recently demonstrated that high
$Q$-factors at visible wavelengths can be achieved when embedding a high-
quality diamond membrane into the cavity [36]. In comparison to ring
resonators, there are no limitations due to bending losses and surface
roughness is less detrimental.

##  II Methods

Our plano-concave microcavity design supports Gaussian fundamental modes (Fig.
1 (a)) [37, 38, 39, 40, 41, 42]. The microcavity is formed by two mirror-
coated fused silica substrates, one of which contains an array of spherical
micromirrors fabricated via laser ablation [43] enabling efficient coupling to
a single free-space mode [33, 44]. The radii of curvature of these
micromirrors are $\sim 10\,\upmu\textrm{m}$ resulting in a beam waist of $\sim
1\,\upmu\textrm{m}$. We integrate high-quality single-crystalline diamond
micromembranes ($\sim 20\times 20\times 0.8\,\upmu\textrm{m}^{3}$) into the
cavity using a micromanipulator [45, 46] (see Appendix .1).

For conventional Fabry-Perot resonators, the resonance wavelength changes
linearly with the mirror separation $t_{\textrm{a}}$. However, the presence of
a diamond membrane with thickness $t_{\textrm{d}}$ significantly alters this
linear mode structure. The hybridization of modes confined in the air and
diamond layers of the resonator manifests in avoided crossings (Fig. 1 (b))
[46, 41, 47, 48, 49, 50]. The cavity resonance frequencies depend on both the
separation of the two mirrors and the thickness of the diamond at the location
of the cavity mode (see Appendix .2).

One of the main advantages of our cavity platform is the in situ tuning
capability. Both the separation of the mirrors and the lateral position of the
cavity mode with respect to the diamond membrane can be controlled via a stack
of piezoelectric nano-positioners (attocube ANPx51, ANPz51). By including a
slight thickness gradient into the diamond membrane, we are able to tune the
exact diamond thickness of the membrane in the cavity by adjusting the
relative lateral position of the mirrors. The in situ tuning capability allows
us to control both the absolute frequency as well as the relative splitting of
the resonator modes.

Excitation of the first order Stokes process in diamond can be modelled as a
three-level atom-like system (inset Fig. 1 (c)) involving a ground state
$\left|1\right>$, a virtual state $\left|2\right>$ and a meta-stable state
$\left|3\right>$. A pump laser excites the ground-state population from
$\left|1\right>$ to $\left|2\right>$. The system decays via state
$\left|3\right>$ emitting a red-shifted photon
($\left|2\right>\rightarrow\left|3\right>$) followed by an optical phonon of
fixed frequency ($\sim 40\,$THz, $\left|3\right>\rightarrow\left|1\right>$).

By coupling both the pump and the Stokes photons to a cavity mode, the Stokes
process can be strongly enhanced [51]. It should be noted that no population
inversion is required for stimulated Raman scattering and hence the creation
of a Raman laser [52]. Importantly, the gain of the Raman process is maximized
by strong confinement of and coupling between the pump and Raman modes, which
suggests the use of fundamental resonator modes. Careful tuning of the mirror
separation $t_{\textrm{a}}$ and the diamond thickness $t_{\textrm{d}}$ allows
the double-resonance condition to be established for a wide range of pump
wavelengths in the visible wavelength regime. When changing the pump
wavelength $t_{\textrm{d}}$ and $t_{\textrm{a}}$ need to be adjusted such that
the cavity both remains resonant with the pump laser while another mode is
red-detuned exactly by the Raman shift (see Appendix .3).

The frequency of the Raman output is determined by the frequency of the pump
laser and the fixed Raman shift. We use a continuous-wave (cw) narrow-band
tunable red diode laser (Toptica DL Pro 635, $\lambda=630\dots
640\,\textrm{nm}$) as a pump source. The operation wavelength range of the
cavity is given by the reflective stopband of the DBR, which we determine
using a white-light transmission measurement [33]. Figure 1 (c) displays the
stopband of the planar bottom mirror which is centered around
$\lambda_{\textrm{c,bot}}=625\,\textrm{nm}$; the reflectivity is more than 99%
over a bandwidth of $\sim 100$ THz. The top mirror has similar properties but
with a stopband centered at $\lambda_{\textrm{c,top}}=629\,\textrm{nm}$.

##  III Results

To characterize the mode structure of the cavity, we couple a cw green laser
at $532\,\textrm{nm}$ into the cavity through the curved top mirror. We tune
the mirror separation by applying a voltage to the piezo using a highly-stable
voltage source (Basel Precision Instruments SP 927). Background
photoluminescence (PL) from the diamond acts as an internal light source and
couples to the different resonator modes [33]. Figure 1 (c) displays a PL
spectrum collected through the top mirror. We set the mirror separation such
that the splitting between the modes with effective mode numbers
$q_{\textrm{eff}}=17$ and $q_{\textrm{eff}}=19$ corresponds to the Raman shift
in diamond ($\Delta\nu_{\textrm{R}}\sim 40\,\textrm{THz}$). We define the
effective mode number by the number of half wavelengths between the two
mirrors, i.e. within the air-gap and the diamond layer,
$q_{\textrm{eff}}\lambda/2\approx
t_{\textrm{a}}+n_{\textrm{d}}t_{\textrm{d}}$. A small deviation from integer
values of $q_{\textrm{eff}}$ is caused by field penetration into the DBR
mirrors [53, 54]. Using a one-dimensional transfer-matrix calculation we infer
the mirror separation and diamond layer thickness to be
$t_{\textrm{a}}=4.18\,\upmu\textrm{m}$ and $t_{\textrm{d}}=756\,\textrm{nm}$,
respectively (Figure 1 (b), Appendix .2).

Next, we verify that we are able to establish the double resonance condition
by coupling an additional laser resonant with mode $q_{\textrm{eff}}=17$ at
${\lambda_{\textrm{pump}}=632.99\,\textrm{nm}}$ into the cavity. The resulting
Raman scattered light is at a wavelength of
$\lambda_{\textrm{R}}$=(1/$\lambda_{\textrm{pump}}-\Delta\nu_{\textrm{R}}/c)^{-1}=692.25\,\textrm{nm}$,
where $c$ is the speed of light. We then tune the mirror separation and record
spectra from the cavity (Fig. 2 (a)). As expected, the cavity modes with
effective mode numbers $q_{\textrm{eff}}=16...18$ and wavelengths in the range
of $\lambda=670...700\,\textrm{nm}$ redshift with increasing mirror
separation. When the pump laser at
$\lambda_{\textrm{pump}}=632.99\,\textrm{nm}$ is resonant with the
$q_{\textrm{eff}}=18...20$ modes for relative mirror separations of
$-316.5\,\textrm{nm}$, $0\,\textrm{nm}$ and $+316.5\,\textrm{nm}$, narrow
peaks appear in the spectrum at CCD pixels corresponding to
$\lambda=691.19\,\textrm{nm}$ and $\lambda=691.32\,\textrm{nm}$, as
highlighted in the insets of Fig. 2 (a).

The linecut at $\lambda=691.19\,\textrm{nm}$ clearly shows that the cavity
resonances for $q_{\textrm{eff}}=16$ and $q_{\textrm{eff}}=18$ appear at
smaller ($-345.5\,\textrm{nm}$) and larger mirror separations
($+345.5\,\textrm{nm}$) than the Raman peaks, respectively. Only for a mirror
separation of $t_{\textrm{a}}=4.18\,\upmu\textrm{m}$ $\lambda_{\textrm{pump}}$
and $\lambda_{\textrm{R}}$ are, within the spectrometer resolution,
simultaneously resonant with the cavity for $q_{\textrm{eff}}=19$ and
$q_{\textrm{eff}}=17$. For this double resonance condition the signal
intensity is increased by over three orders of magnitude compared to the other
peaks.

In the following we denote the wavelength of the cavity mode with
$q_{\textrm{eff}}=17$ close in wavelength to that of the pump
($\lambda_{\textrm{pump}}$) as $\lambda_{\textrm{p}}^{\textrm{cav}}$; and the
wavelength of the cavity mode with $q_{\textrm{eff}}=19$ close in wavelength
to that of the Raman photon ($\lambda_{\textrm{R}}$) as
$\lambda_{\textrm{S}}^{\textrm{cav}}$. An analogous notation is adapted for
the corresponding frequencies $\nu$.

A faster way to confirm that the double resonance condition is satisfied is
displayed in Fig. 2 (b). Here, we only couple the diode laser at
$\lambda_{\textrm{pump}}$ into the cavity and record the cavity transmission
using a photodiode located beneath the bottom mirror. The transmission
spectrum reveals several peaks at mirror separations where the pump laser is
resonant with the cavity. These peaks are associated with fundamental and
higher order cavity modes. Simultaneously, we measure the cavity emission at
wavelengths $>644\textrm{nm}$ using a single photon counting module. A strong
signal is observed only when $\lambda_{\textrm{pump}}$ is resonant with mode
$q_{\textrm{eff}}=19$, while at the same time $\lambda_{\textrm{R}}$ is
resonant with $q_{\textrm{eff}}=17$. The correlation between a peak in
transmission (signifying a resonant pump laser) and a strong peak in cavity
emission at longer wavelengths (signifying a resonant Raman process) is a
clear demonstration that the double resonance condition is satisfied.

Figure 2: Demonstration of cavity-enhanced doubly-resonant Raman scattering.
(a) (top) Optical spectra as a function of relative cavity length reveal the
cavity mode structure. Here, a laser at $532\,\textrm{nm}$ excites background
PL in the diamond and a narrowband diode laser at
$\lambda_{\textrm{pump}}=632.99\,\textrm{nm}$ drives Raman scattering. For
$q_{\textrm{eff}}=19$ and $q_{\textrm{eff}}=17$, the pump and Raman scattered
light are resonant simultaneously, i.e. the double resonance condition is met.
(bottom) Linecut at $\lambda=691.19\,\textrm{nm}$ highlighting the strong
signal enhancement of doubly-resonant Raman scattering. (b) Cavity
transmission (top) and cavity emission at wavelengths $>644\textrm{nm}$
measured with a single photon counter (bottom) as a function of cavity length
for $\lambda_{\textrm{pump}}=632.99\,\textrm{nm}$. A cavity signal is only
observed when the double resonance condition is satisfied. The piezo voltage
is changed at a rate of $2\,\textrm{mV}/\textrm{ms}$. (c) $Q$-factor
measurement for
$\nu_{\textrm{p}}^{\textrm{cav}}=c/\lambda_{\textrm{p}}^{\textrm{cav}}=473.233\,\textrm{THz}$.
Sidebands created by an EOM at $\nu_{\textrm{pump}}\pm 3.9\,\textrm{GHz}$ act
as a frequency ruler to extract the cavity linewidth. (d) $Q$-factor as a
function of cavity resonance frequency.

Next, we determine the quality factor of the pump mode of the cavity,
$Q_{\textrm{p}}$, following the method reported in Ref. [36]. To extract the
cavity linewidth, we keep the laser frequency $\nu_{\textrm{pump}}$ fixed
while scanning the cavity length, monitoring the reflected light on a
photodiode. An electro-optic modulator (EOM, Jenoptik PM635) is used to create
laser side-bands at $\nu_{\textrm{pump}}\pm 3.9\,\textrm{GHz}$, thereby
providing a frequency ruler to extract the cavity linewidth. Fig.2 (c) shows
the reflected signal averaged over 200 scans for
$\nu_{\textrm{pump}}=473.233\,\textrm{THz}$. Assuming a linear response of the
piezo across the resonance, we extract a cavity mode full width at half
maximum (FWHM) linewidth of $\delta\nu_{\textrm{p}}^{\textrm{cav}}=(1.59\pm
0.05)\,\textrm{GHz}$ corresponding to a quality factor of
$Q_{\textrm{p}}=297\,000\pm 500$. Fig. 2 (d) shows the dependence of
$Q_{\textrm{p}}$ on the cavity resonance frequency
$\nu_{\textrm{p}}^{\textrm{cav}}$.

Figure 3: (a) Series of cavity spectra for different detunings of the pump
laser from the double resonance condition
$\nu_{\textrm{pump}}-\nu_{\textrm{pump,dres}}$. We ensure that the pump laser
is resonant with the cavity for every spectrum by continuously modulating the
cavity length. The cavity spectra comprise the Raman signal at
$\nu_{\textrm{R}}$ and the closest cavity mode at
$\nu_{\textrm{S}}^{\textrm{cav}}$. We fit the amplitude of the Raman signal
for every value of $\nu_{\textrm{pump}}-\nu_{\textrm{pump,dres}}$ with a
Lorentzian (projected blue points) with linewidth of 519.8 GHz. The center
frequency of the fit indicates that the double resonance condition is
satisfied when $\nu_{\textrm{R,dres}}=432.520\,\textrm{THz}$ for
$\nu_{\textrm{pump,dres}}=472.434\,\textrm{THz}$. (b) The individual cavity
spectra can be fitted well using a model based on two multiplied Lorentzians
centered at the Stokes frequency $\nu_{\textrm{R}}-\nu_{\textrm{R,dres}}$ and
the frequency of the adjacent cavity mode
$\nu_{\textrm{S}}^{\textrm{cav}}-\nu_{\textrm{R,dres}}$. The inset in each
panel indicates the pump frequency
$\nu_{\textrm{pump}}-\nu_{\textrm{pump,dres}}$.

Next, we characterize in more detail the exact detuning dependence of the
double resonance condition by tuning the pump laser. We exploit the fact that
Raman signal in the cavity is only generated when a cavity-mode is resonant
with the pump laser, $\nu_{\textrm{pump}}=\nu_{\textrm{p}}^{\textrm{cav}}$. In
this event, two signals will appear in the spectra: Raman scattering of the
pump laser in the cavity at
$\nu_{\textrm{R}}=\nu_{\textrm{pump}}+\Delta\nu_{\textrm{R}}$, and the cavity
mode nearest in frequency to this Raman signal,
$\nu_{\textrm{S}}^{\textrm{cav}}$, which is fed by PL light that is non-
resonantly generated by the pump laser. Using this technique it is not
necessary to keep the cavity at one particular resonance – this circumvents
any problems caused either by drift or acoustic and thermal noise.

We modulate the cavity length continuously and record the resulting cavity
spectra (Fig. 3 (a)) varying
$\nu_{\textrm{pump}}=\nu_{\textrm{p}}^{\textrm{cav}}$
($\lambda_{\textrm{pump}}=\lambda_{\textrm{p}}^{\textrm{cav}}$) from 468.475
THz (639.932 nm) to 474.471 THz (631.845 nm). We find that for the pump
frequency $\nu_{\textrm{pump,dres}}=472.434\,\textrm{THz}$ the double
resonance condition is fulfilled,
$\nu_{\textrm{R}}=\nu_{\textrm{S}}^{\textrm{cav}}=\nu_{\textrm{R,dres}}=432.520\,\textrm{THz}$.
These values are different from those in Fig. 2 due to a slightly different
lateral position of the cavity mode corresponding to a different diamond
thickness. We determine a Raman shift of
$\Delta\tilde{\nu}_{\textrm{R}}=\nu_{\textrm{pump,dres}}/c-\nu_{\textrm{R,dres}}/c=1331.4\,\textrm{cm}^{-1}$
($\Delta\nu_{\textrm{R}}=c\Delta\tilde{\nu}_{\textrm{R}}=39.914\,\textrm{THz}$),
in good agreement with the previously reported value, $\sim
1332\,\textrm{cm}^{-1}$ [55, 56, 5]. We plot the peak Raman counts for
different detunings of the pump laser from the double resonance condition,
$\nu_{\textrm{pump}}-\nu_{\textrm{pump,dres}}$ (projected blue points in Fig.
3 (a)). We find that these peak counts follow a Lorentzian with FWHM linewidth
of 519.8 GHz [51]. The corresponding projected Raman amplitude is fitted well
by a Lorentzian with FWHM linewidth of 502.9 GHz (projected red points in Fig.
3 (a)).

We fit the individual spectra for different detunings of the pump laser (with
respect to the double-resonance frequency $\nu_{\textrm{pump,dres}}$) to the
product of two Lorentzians describing the cavity mode at
$\nu_{\textrm{S}}^{\textrm{cav}}$ (FWHM
$\delta\nu_{\textrm{S}}^{\textrm{cav}}$) and the gain bandwidth of the Raman
scattering process at $\nu_{\textrm{R}}$ (FWHM $\delta\nu_{\textrm{R}}$).
These fits allow the peak positions and linewidths to be extracted (Fig. 3)
[33]. Figures 4 (a,b) display the results of these fits. Over the tuning range
of the pump laser, the detuning between $\nu_{\textrm{S}}^{\textrm{cav}}$ and
$\nu_{\textrm{R}}$ varies from $-319.7\,\textrm{GHz}$ to
$526.7\,\textrm{GHz}$. The linewidth of the Raman gain for the different fits
is $\delta\nu_{\textrm{R}}=(48.3\pm 1.6)$ GHz corresponding to
$Q_{\textrm{R}}=8\,960\pm 290$. This Raman linewidth agrees well with
previously reported values (40.8…47.8 GHz) [57, 58, 33], indicating low strain
in the diamond membrane. The linewidth of the cavity mode closest to
$\nu_{\textrm{R}}$ at $\nu_{\textrm{S}}^{\textrm{cav}}$ decreases from
$\delta\nu_{\textrm{S}}^{\textrm{cav}}=(167.3\pm 0.8)\,\textrm{GHz}$ to
$(47.0\pm 0.4)\,\textrm{GHz}$ for increasing
$\nu_{\textrm{pump}}=\nu_{\textrm{p}}^{\textrm{cav}}$, which is expected from
the increase in reflectivity on approaching the stopband center of the DBR
mirror coatings. The corresponding $Q$-factor increases from
$Q_{\textrm{p}}=2\,570\pm 90$ to $9\,250\pm 90$. At the double-resonance
condition, the $Q$-factor of the Stokes cavity mode is
$Q_{\textrm{S,dres}}=6\,650\pm 50$.

Figure 4: (a) Peak position of the cavity ($\nu_{\textrm{S}}^{\textrm{cav}}$)
and Raman scattered light ($\nu_{\textrm{R}}$) as a function of pump frequency
$\nu_{\textrm{pump}}$ relative to the double resonance condition at
$\nu_{\textrm{pump,dres}}$ and $\nu_{\textrm{R,dres}}$. (b) Linewidth of the
cavity ($\delta\nu_{\textrm{S}}^{\textrm{cav}}$) and the Raman gain
($\delta\nu_{\textrm{R}}$) versus pump frequency
$\nu_{\textrm{pump}}-\nu_{\textrm{pump,dres}}$. (c) Spectrally resolved cavity
signal with increased pump power at the double-resonance condition. (d)
Integrated peak intensity of peaks in (c) as a function of pump power. The
linear behavior suggests that no lasing occurs.

Next, we perform double-resonance measurements for different pump powers
$P_{\textrm{pump}}$ (as measured before the sample objective) (Fig. 4 (c,d)).
Up to the largest available pump power in the experiment, the intensity
increases linearly: there is no superlinear dependence presaging Raman lasing.

To estimate the threshold power required to establish Raman lasing, we analyze
the performance of the cavity. Raman scattering in a microcavity can be
described using classical coupled mode equations [59, 60]. Lasing occurs when
the round-trip gain equals the round-trip loss. Assuming that both the pump
laser and the Raman light are resonant with the cavity,
$\lambda_{\textrm{pump}}=\lambda_{\textrm{p}}^{\textrm{cav}}$ and
$\lambda_{\textrm{R}}=\lambda_{\textrm{S}}^{\textrm{cav}}$, the lasing
threshold $P_{\textrm{th}}$ can be calculated via (see Appendix .5):

$P_{\textrm{th}}=\frac{1}{\eta}\frac{2n_{\textrm{S}}n_{\textrm{p}}\pi^{2}}{\lambda_{\textrm{S}}^{\textrm{cav}}\lambda_{\textrm{p}}^{\textrm{cav}}g_{\textrm{R}}^{\textrm{B}}}\frac{V_{\textrm{R}}(Q_{\textrm{S}}+Q_{\textrm{R}})}{Q_{\textrm{S}}^{2}Q_{\textrm{p}}}\,.$
(1)

Here, $\lambda_{\textrm{p(S)}}^{\textrm{cav}}$, $n_{\textrm{p(S)}}$ and
$Q_{\textrm{p(S)}}$ are the wavelengths, refractive indices and $Q$-factors
for the cavity modes resonant with the pump laser and the Raman light,
respectively. $Q_{\textrm{R}}$ is the quality factor corresponding to the
bandwidth of the Raman gain. The bulk Raman gain coefficient in the employed
pump wavelength range is $g_{\textrm{R}}^{\textrm{B}}\sim 40\,\textrm{cm/GW}$
[61]. The power incoupling efficiency $\eta$ can be extracted from the cavity
transmission measurement displayed in Fig. 2 (c) [62, 63]. From the dip in
reflection, we infer a power incoupling efficiency of
$\eta_{\text{C}}=1-P_{\textrm{R}}/P_{\textrm{0}}=0.45$.

Modeling the cavity using a one-dimensional transfer-matrix model along with
Gaussian optics, we estimate a Raman mode volume of
$V_{\textrm{R}}=109.85\,\upmu\textrm{m}^{3}$ (see Appendix .4). Taking
$n_{\textrm{p(S)}}=n_{\textrm{dia}}=2.4$, we find
$P_{\textrm{th}}=187.50\,\textrm{mW}$. This relatively low threshold power
constitutes a reduction in threshold power by more than an order of magnitude
with respect to a bulk Raman laser in the visible [8, 9, 64] With realistic
improvements we predict that our device platform could feature threshold
powers in the sub-mW range (see Appendix .6).

Figure 5: (a) Demonstration of the tunability of the double resonance
condition. Owing to the thickness gradient, shifting the diamond laterally
changes the diamond thickness in the cavity and subsequently the condition for
double resonance. (b) The diamond thickness and the width of the air-gap are
extracted from one-dimensional transfer-matrix simulations. The double
resonance is condition is satisfied when the pump mode (blue) and the Stokes
mode (red) cross. For
$\lambda_{\textrm{p}}^{\textrm{cav}}=633.52\,\textrm{nm}$ and
$\lambda_{\textrm{S}}^{\textrm{cav}}=691.90\,\textrm{nm}$, we extract
$t_{\textrm{d}}=754.58\,\textrm{nm}$ and
$t_{\textrm{a}}=4173.98\,\textrm{nm}$. (c) Lateral displacement of the cavity
mode plotted against diamond thickness $t_{\textrm{d}}$. Here, the relative
position of the cavity mode is calculated with respect to the corners of the
diamond. Extracting the diamond thickness from the double-resonance
measurements in (a) gives a thickness gradient of
$0.17\,\textrm{nm}/\upmu\textrm{m}$. (d) Linear shift of the double-resonance
condition with the diamond thickness resulting in a red-shift of the Stokes
wavelength.

Next, we demonstrate the possibility to tune the double resonance condition by
changing the thickness of the diamond layer within the cavity mode in situ
(Fig. 5 (a)). To this end, we laterally displace the cavity mode with respect
to the diamond membrane, exploiting a small thickness gradient (Fig. 1 (a)).
Over the lateral fine-tuning range of the nanopositioner (travel range $\sim
4\,\upmu\textrm{m}$), the double resonance condition can be tuned from
$\nu_{\textrm{pump,dres}}=471.44\,\textrm{THz}$ to $472.29\,\textrm{THz}$
($\nu_{\textrm{R,dres}}=431.52\,\textrm{THz}$ to $432.37\,\textrm{THz}$), a
continuous tuning range of 0.85 THz. Considering that the width of the double
resonance gain profile is $>$500 GHz, this would enable a $>$THz continuous
tuning range of the lasing frequency.

To extract the exact diamond thickness, we perform one-dimensional transfer-
matrix-based simulations of the cavity mode structure (see Appendix .2). For
these simulations, we use the exact mirror structure obtained from fitting the
mirror stopband (Fig. 1 (c) [33, 36]), and sweep the width of the air-gap
$t_{\textrm{a}}$ and the diamond thickness $t_{\textrm{d}}$ for fixed
wavelengths $\lambda_{\textrm{p}}^{\textrm{cav}}$ and
$\lambda_{\textrm{S}}^{\textrm{cav}}$. The double resonance condition is met
whenever the modes for $\lambda_{\textrm{p}}^{\textrm{cav}}$ cross the modes
for $\lambda_{\textrm{S}}^{\textrm{cav}}$. Fig. 5 (b) shows a transfer-matrix
simulation for $\lambda_{\textrm{p}}^{\textrm{cav}}=633.52\,\textrm{nm}$ and
$\lambda_{\textrm{S}}^{\textrm{cav}}=691.90\,\textrm{nm}$ (extracted from Fig.
4 (a)). Here, the two cavity modes overlap for
$t_{\textrm{a}}=4173.98\,\textrm{nm}$ and
$t_{\textrm{d}}=754.85\,\textrm{nm}$.

We extract $t_{\textrm{d}}$ for all measurements displayed in Fig. 5 (a) and
plot $t_{\textrm{d}}$ versus lateral displacement of the cavity mode (Fig. 5
(c)). To calibrate the lateral displacement, we use the edges of the diamond
($\sim 18\,\upmu\textrm{m}$) measured with a laser scanning confocal
microscope (Keyence Corporation) as a reference. We find a thickness gradient
$|\frac{\Delta t_{\textrm{d}}}{\Delta x}|=\left(0.17\pm
0.2\right)\,\textrm{nm}/\upmu\textrm{m}$. As shown in Fig. 5 (d), we observe a
linear shift of $\lambda_{\textrm{S}}^{\textrm{cav}}$ with $t_{\textrm{d}}$.
From our simulations, we find that for the right combination of
$t_{\textrm{d}}$ and $t_{\textrm{a}}$, the double resonance condition can be
tuned continuously across the whole mirror stopband corresponding to a
continuous tuning range of tens of THz (see. Appendix .3).

##  IV Conclusion

In conclusion, we demonstrate a platform for the widely-tunable doubly-
resonant enhancement of Raman scattering from diamond based on a tunable open-
access microcavity. The in situ tuning capability of our device provides a
convenient way to establish a double resonance condition in which both pump
and Raman wavelengths are resonant with a cavity mode. Exploiting a slight
thickness gradient of the incorporated diamond membrane enables the doubly-
resonant configuration to be achieved over a wide tuning range of more than 1
THz. These results, together with the high quality factors of the cavity in
the visible wavelength range, suggest that Raman lasing can be achieved with
the present system. We predict a lasing threshold of $188\,\textrm{mW}$, a
reduction by more than an order of magnitude compared to bulk Raman lasers
[4]. We anticipate that with realistic improvements of our platform, sub-mW
Raman lasing thresholds can be achieved. Importantly, we predict that there
are configurations where mode-hop-free tuning of the double resonance
condition over tens of THz is possible, in principle limited only by the
spectral width of the reflective stopband of the mirrors. These advancements
pave the way to a universal, low-power, frequency-shifter. Finally, we note
that due to the generic design of our platform, other wide-bandgap Raman laser
materials such as aluminum nitride [65] can readily be incorporated into our
device. A wider point is that the integration of materials exhibiting a strong
$\chi^{(2)}$ nonlinearity such as silicon carbide [66, 67], lithium niobate
[68] or gallium phosphide [69, 70] could enable low-threshold frequency
conversion using other nonlinear processes, for instance second-harmonic
generation or sum- and difference-frequency mixing.

##  V Acknowledgments

We acknowledge financial support from NCCR QSIT, a competence center funded by
SNF; the Swiss Nanoscience Institute (SNI); ITN networks S3NANO and SpinNANO;
and SNF Grant No. 200021_143697. DR acknowledges support from the Swiss
National Science Foundation (Project P400P2_194424). SF and DR contributed
equally to performing the experiments, analyzing the data and development of
the theory.

## APPENDIX

###  .1 Methods

The core of this experiment is the tunable, planar-concave Fabry-Perot
microcavity [40, 39] with an embedded diamond micromembrane, depicted
schematically in Fig. 1 (a). The microcavity comprises two fused silica
substrates exhibiting highly reflective dielectric mirror coatings (ECI
evapcoat). Prior to applying the coating, we fabricate an array of spherical
micro-indentations via CO2 laser ablation [43] in one of the substrates. The
micro-indentations feature small radii of curvature ${R_{\textrm{cav}}\sim
10\,\upmu\textrm{m}}$ and depths $d\sim 1.5\,\upmu\textrm{m}$. We employ 14
(15) $\lambda_{\textrm{c}}/4$ layers of a SiO2/Ta2O5 distributed Bragg
reflector for the curved top (planar bottom) mirrors. From a white-light
transmission measurement [33, 36, 71], the center of the stopband of the top
mirror is determined to be $\lambda_{\textrm{c}}=625\,\textrm{nm}$ (Fig. 1
(c)). Using a transfer-matrix-based refinement algorithm (Essential Macleod)
we can reconstruct the reflection spectrum utilizing an individual layer-
thickness tolerance of 3% with $n_{\textrm{SiO}_{2}}=1.46$ and
$n_{\textrm{Ta}_{2}\textrm{O}_{5}}=2.11$. Using the same approach we find
$\lambda_{\textrm{c}}=629\,\textrm{nm}$ for the top mirror.

Starting with commercially available high-purity, $\langle$100$\rangle$-cut
single crystal diamond (Element Six), we fabricate membranes via inductively-
coupled reactive-ion etching and electron-beam lithography [72, 45, 73]. We
then transfer membranes with typical dimensions $\sim 20\times 20\times
0.8\,\upmu\textrm{m}^{3}$ to the planar mirror using a micromanipulator [45,
46]. The diamond membranes exhibit a slight thickness gradient introduced
during the thinning of the diamond [74].

The bottom mirror is mounted on a stack of xyz-piezoelectric nanopositioners
(attocube, 2$\times$ANPx51 and ANPz51) and placed inside a homebuilt titanium
“cage”; the top mirror is rigidly attached to the top of the cage [36]. By
applying a voltage to the nanopositioners, the bottom mirror can be moved in
all three dimensions with respect to the top mirror, offering both spatial and
spectral tunability [46]. Finally, the titanium cage is mounted on top of a
high-precision mechanical stage (Newport, M-562-XYZ) to enable the cavity
output to be coupled to external detection optics [33].

We use a narrow-band tunable red diode-laser as pump laser (Toptica DL Pro
635, $\lambda_{\textrm{pump}}=630\cdots 640\,\textrm{nm}$, $\delta\nu\leq
500\,\textrm{kHz}$). This pump laser is spectrally filtered (Semrock,
FF01-637/7-25 and FF01-650/SP-25) and then coupled into the cavity using an
objective of moderate numerical aperture (Microthek, 20$\times$, NA=0.4) [33].
The Stokes signal is collected via the same objective in a back-scattering
geometry (Fig. 1 (a)). A combination of a dichroic mirror (cutoff 644 nm, AHF
F48-644) and a long-pass filter (Semrock, BLP02-635R-25) is used to filter the
excitation laser from the signal. The Stokes signal is then coupled into a
single-mode detection fiber (Thorlabs, 630-HP) and recorded with a
spectrometer.

###  .2 Cavity mode structure

Conceptually, the cavity mode can be described using a coupled two-cavity
model: one cavity is confined to the diamond bound by the bottom DBR and the
diamond-air interface; the other cavity is confined to the air bounded by the
diamond-air interface and the top DBR [33, 36, 75]. Figure 6 displays a one-
dimensional transfer-matrix calculation of the cavity mode structure, using
the mirror structure extracted from Fig. 1 (c). These calculations confirm
that the locations of the avoided crossings in the mode structure depend on
the choice of diamond thickness $t_{\textrm{d}}$. We find that for
$t_{\textrm{a}}=4173.98\,\textrm{nm}$ and
$t_{\textrm{d}}=754.58\,\textrm{nm}$, the spectrum observed in Fig. 1 (c) is
reproduced well.

Figure 6: One-dimensional transfer-matrix calculation of the cavity mode
structure, i. e. resonance frequencies for different mirror separations
$t_{\textrm{a}}$, for different diamond thicknesses $t_{\textrm{d}}$. The non-
linear mode dispersion arises as a consequence of hybridization of cavity
modes resonant with the air-gap (indicated by the dashed burgundy line) and
the diamond (dashed green line), respectively. The hybridization leads to
avoided crossings whose position depends on the exact diamond thickness. The
central panel shows the mode-structure for the extracted diamond thickness
($t_{\textrm{d}}\simeq 755\,\textrm{nm}$) from Fig. 5 (b). The orange circles
indicate a pair of cavity modes whose frequencies are separated by the Raman
shift of $\Delta\nu_{\textrm{R}}\sim 40\,\textrm{THz}$ (black arrow).

###  .3 Tuning of double resonance condition

Open-access microcavities offer a convenient tuning mechanism of their
resonance frequency simply by changing the separation of the two mirrors
($t_{\textrm{a}}$) using a piezoelectric nanopositioner. Importantly, such
cavities offer another tuning mechanism where, rather than the width of the
air-gap, the thickness of the material layer is changed in situ. Here, a small
thickness gradient in the diamond membrane converts a lateral displacement of
the cavity mode into a change of the membrane thickness ($t_{\textrm{d}}$).
Tuning both $t_{\textrm{a}}$ and $t_{\textrm{d}}$ allows both the absolute
wavelength and the spacing of the cavity modes to be controlled. As a
consequence, a gradient in the diamond thickness $|\frac{\Delta
t_{\textrm{d}}}{\Delta x}|$ enables the double resonance condition to be
satisfied for different pairs of wavelengths. In Fig. 5 (a) we demonstrate
experimentally a continuous tuning range of the double resonance condition by
0.85 THz. This is achieved by changing the diamond thickness by $\sim
0.6\,\textrm{nm}$, from 755.5 nm to 756.2 nm.

To explore this tuning mechanism in more detail, we perform one-dimensional
transfer-matrix calculations (Essential Macleod). We calculate the
combinations of air-gap width $t_{\textrm{a}}$ and diamond thickness
$t_{\textrm{d}}$ at which specific wavelengths are resonant. We perform these
calculations for cavity wavelengths within the tuning range of our pump laser,
i.e. $\lambda_{\textrm{p}}^{\textrm{cav}}=630\cdots 640\,\textrm{nm}$ and for
a range of $t_{\textrm{a}}$ and $t_{\textrm{d}}$ accessible with the device
presented in the main text (solid lines Fig. 7 (a)). We then add calculations
for the corresponding wavelengths red-shifted by the Raman shift at
$\lambda_{\textrm{S}}^{\textrm{cav}}=(1/\lambda_{\textrm{p}}^{\textrm{cav}}-\Delta\nu_{\textrm{R}}/c)^{-1}$
(dashed lines Fig. 7 (a)). At pairs of $t_{\textrm{a}}$ and $t_{\textrm{d}}$
where the solid and dashed line cross, the double-resonance condition is
satisfied.

We find that, in principle, the double resonance condition can be tuned
continuously from
$\lambda_{\textrm{pump,dres}}=625.00\,\textrm{nm}(\lambda_{\textrm{R,dres}}=681.75\,\textrm{nm})$
to
$\lambda_{\textrm{pump,dres}}=649.00\,\textrm{nm}(\lambda_{\textrm{R,dres}}=710.41\,\textrm{nm})$
($17.3\,\textrm{THz}$) by changing the diamond thickness from 751.4 nm to
763.8 nm (green points in Fig. 7 (a)). The experimentally demonstrated tuning
range is indicated by the purple points in Fig. 7 (a).

By optimizing the choice of $t_{\textrm{a,d}}$, we find a configuration which
in principle allows the double-resonance condition to be tuned in a mode-hope
free fashion harnessing the whole stopband of the mirror (72.2 THz,
$\lambda_{\textrm{pump,dres}}=565\cdots 645\,\textrm{nm}$,
$\lambda_{\textrm{R,dres}}=610.98\cdots 705.62\,\textrm{nm}$, see Fig. 7 (b)).

Figure 7: One-dimensional transfer-matrix calculations of the cavity
resonances as a function of $t_{\textrm{a,d}}$ for different combinations of
$\lambda_{\textrm{p}}^{\textrm{cav}}$ (solid line) and
$\lambda_{\textrm{S}}^{\textrm{cav}}$=(1/$\lambda_{\textrm{p}}^{\textrm{cav}}-\Delta\nu_{\textrm{R}}/c)^{-1}$
(dashed line) at specific wavelengths. The double resonance condition is
satisfied when the corresponding solid and dashed lines cross (indicated by
same color). By changing the diamond thickness, the condition for the double-
resonance can be tuned continuously, as indicated by the green circles. (a)
Calculations for pairs of $t_{\textrm{a,d}}$ in a range accessible with the
device presented in the main text. The calculations suggest a continuous
tuning range of $17.3\,\textrm{THz}$. The experimentally verified tuning is
indicated by the purple circles. (b) Optimizing the choice of
$t_{\textrm{a,d}}$ enables continuous tuning across the entire reflective
stopband, amounting to $72.2\,\textrm{THz}$. For simplicity, only cavity modes
for which continuous tuning is possible are included in (b).

###  .4 Calculation of effective Raman mode volume

We consider a doubly-resonant system
($\omega_{\textrm{p}}=\omega_{\textrm{pump}}=\omega_{\textrm{p}}^{\textrm{cav}}$
and
$\omega_{\textrm{S}}=\omega_{\textrm{R}}=\omega_{\textrm{S}}^{\textrm{cav}}$).
In the following, we omit the ”cav” superscripts for concise notation and
clarity. The effective Raman mode volume accounts for the spatial overlap of
the pump (p) and Stokes (S) cavity modes and can be determined via [59, 76]

$V_{\textrm{R}}=\frac{\int_{\textrm{cav}}n^{2}_{\textrm{p}}(\vec{r})|\vec{E}_{\textrm{p}}(\vec{r})|^{2}\textrm{d}^{3}r\times\int_{\textrm{cav}}n^{2}_{\textrm{S}}(\vec{r})|\vec{E}_{\textrm{S}}(\vec{r})|^{2}\textrm{d}^{3}r}{\int_{\textrm{dia}}n^{2}_{\textrm{p}}(\vec{r})|\vec{E}_{\textrm{p}}(\vec{r})|^{2}\times
n^{2}_{\textrm{S}}(\vec{r})|\vec{E}_{\textrm{S}}(\vec{r})|^{2}\textrm{d}^{3}r}\,,$
(2)

where $\vec{E}_{p(S)}(\vec{r})$ is the pump (Stokes) electric field at
position $\vec{r}$. The integrals over the electric field can be calculated
following the same approach as reported in Ref. [33]. We approximated the beam
waist to be constant $w_{\textrm{0,I}}$ and solve the integral in cylindrical
coordinates:

$\displaystyle\int_{\textrm{cav}}n^{2}(\vec{r})|\vec{E}(\vec{r})|^{2}\textrm{d}^{3}r=$
(3) $\displaystyle=$
$\displaystyle\int_{\textrm{cav}}n^{2}(z)|\vec{E}(z)|^{2}\textrm{d}z\int_{0}^{2\pi}\textrm{d}\phi\int_{0}^{\infty}re^{-r^{2}/2w_{\textrm{I}}^{2}}\textrm{d}r$
$\displaystyle=$ $\displaystyle
2\pi\frac{1}{4}w_{\textrm{I}}^{2}\int_{\textrm{cav}}\epsilon_{\textrm{0}}n^{2}(z)|\vec{E}(z)|^{2}\textrm{d}z\,,$

where $\epsilon_{\textrm{R}}=n^{2}$ and $w_{\textrm{0,I}}$ is the intensity
beam-waist given by [36, 77]

$w_{\textrm{0,I}}=\sqrt{\frac{\lambda}{\pi}}\left[\left(t_{\textrm{a}}+\frac{t_{\textrm{d}}}{n_{\textrm{d}}}\right)\times
R_{\textrm{cav}}-\left(t_{\textrm{a}}+\frac{t_{\textrm{d}}}{n_{\textrm{d}}}\right)^{2}\right]^{\frac{1}{4}}\,.$
(4)

Calculating the respective field profiles according to Eq. 3 reduces Eq. 2 to

$\displaystyle V_{\textrm{R}}$
$\displaystyle=2\pi\frac{1}{4}\left(w_{\textrm{p}}^{2}+w_{\textrm{S}}^{2}\right)\times$
(5)
$\displaystyle\frac{\int_{\textrm{cav}}n^{2}_{\textrm{p}}(z)|E_{\textrm{p}}(z)|^{2}\textrm{d}z\times\int_{\textrm{cav}}n^{2}_{\textrm{S}}(z)|E_{\textrm{S}}(z)|^{2}dz}{\int_{\textrm{dia}}n^{2}_{\textrm{p}}(z)|E_{\textrm{p}}(z)|^{2}\times
n^{2}_{\textrm{S}}(z)|E_{\textrm{S}}(z)|^{2}\textrm{d}z}\,.$

To calculate the Raman mode volume, we approximate the axial vacuum electric-
field distribution with a one-dimensional cavity using a transfer-matrix
calculation (Essential Macleod). We use the exact mirror structure obtained
from fitting the mirror stopband (Fig. 1 (c)) and the combination of
$t_{\textrm{a,d}}$ extracted from Fig. 5 (b). Fig. 8 shows the result of our
calculations. We determine the electric field profile for the pump and Stokes
fields and then their product by numerical integration. Using
$R_{\textrm{cav}}=11\,\upmu\textrm{m}$ and $n_{\textrm{p}}\simeq
n_{\textrm{S}}=2.4$, we calculate the beam waists according to Eq. 4, and find
$w_{\textrm{p}}=1.05\,\upmu\textrm{m}$ and
$w_{\textrm{S}}=1.09\,\upmu\textrm{m}$ taking the
$\lambda_{\textrm{S,p}}^{\textrm{cav}}$ combination extracted from Fig. 4 (a).
Finally, we arrive at $V_{\textrm{R}}=109.85\,\upmu\textrm{m}^{3}$, as quoted
in the main text.

Figure 8: One-dimensional transfer-matrix simulation of the cavity for
$t_{\textrm{a}}=4173.98\,\textrm{nm}$ and
$t_{\textrm{d}}=754.58\,\textrm{nm}$. The top panel shows the refractive index
profile as a function of cavity length. The middle panel shows the profile of
the pump (blue) and Stokes (red) electric field. The bottom panel shows the
overlapped energy density calculated according to the denominator in the
fraction in Eq. 2.

###  .5 Calculation of lasing threshold

To calculate the lasing threshold we follow the approach presented by Checoury
et al. [60, 78]. We consider a doubly-resonant system
($\omega_{\textrm{p}}=\omega_{\textrm{pump}}=\omega_{\textrm{p}}^{\textrm{cav}}$
and
$\omega_{\textrm{S}}=\omega_{\textrm{R}}=\omega_{\textrm{S}}^{\textrm{cav}}$).
In the following, we omit the ”cav” superscripts for concise notation and
clarity. The spacing between the cavity modes is given by
$\omega_{\textrm{S}}=\omega_{\textrm{p}}-\Delta\omega_{\textrm{R}}$ and
$\Delta\omega_{\textrm{R}}$ is the Raman shift. The coupled mode equations
linking the mean Stokes ($N_{\textrm{S}}$) and pump photon numbers
($N_{\textrm{p}}$) are given by:

$\frac{\textrm{d}N_{\textrm{p}}}{\textrm{d}t}=-\frac{N_{\textrm{p}}}{\tau_{\textrm{p}}}-\gamma\frac{N_{\textrm{p}}}{\tau_{\textrm{R}}}-(N_{\textrm{S}}+1)\frac{N_{\textrm{p}}}{\tau_{\textrm{R}}^{\textrm{cav}}}+\kappa_{\textrm{in}}P_{\text{p}}$
(6)
$\frac{\textrm{d}N_{\textrm{S}}}{\textrm{d}t}=-\frac{N_{\textrm{S}}}{\tau_{\textrm{S}}}+(N_{\textrm{S}}+1)\frac{N_{\textrm{p}}}{\tau_{\textrm{R}}^{\textrm{cav}}}\,.$
(7)

Here, $\tau_{\textrm{S}}=Q_{\textrm{S}}/\omega_{\textrm{S}}$ and
$\tau_{\textrm{p}}=Q_{\textrm{p}}/\omega_{\textrm{p}}$ are the Stokes and pump
photon lifetimes. $\gamma$ describes the Raman scattering into modes other
than the cavity mode, and $\tau_{\textrm{R}}$ is a measure for the spontaneous
Raman scattering lifetime in bulk. Stimulated ($N_{\textrm{S}}$) and
spontaneous (+1) Raman scattering into the cavity mode are accelerated with
respect to the bulk scattering rate via Purcell enhancement; the corresponding
lifetime becomes $\tau_{\textrm{R}}^{\textrm{cav}}$. The constant
$\kappa_{\textrm{in}}$ relates the injected pump photon-number per time to the
incident pump power $P_{\text{p}}$.

The spontaneous Raman scattering rate in bulk when the pump mode-polarization
is aligned along the $\langle$110$\rangle$ crystallographic axis can be
calculated via [60]:

$\frac{1}{\tau_{\textrm{R}}}=\frac{2g_{\textrm{R}}^{\textrm{B}}c^{2}\hbar\omega_{\textrm{p}}}{3n_{\textrm{p}}n_{\textrm{S}}V}M\,.$
(8)

Here, $g_{\textrm{R}}^{\textrm{B}}$ denotes the bulk Raman gain and $V$ the
mode volume of a hypothetical large cavity. $M$ characterizes the total number
of Raman modes into which the system can radiate in such a large cavity with
mode volume $V$ for a frequency band of width $\delta\omega_{\textrm{R}}$
[78]:

$M=\frac{V\omega_{\textrm{S}}^{2}n_{\textrm{S}}^{3}}{2\pi
c^{3}}\delta\omega_{\textrm{R}}\,.$ (9)

$\delta\omega_{\textrm{R}}$ describes the FWHM linewidth of the gain profile
of the Raman scattering process. Hence:

$\frac{1}{\tau_{\textrm{R}}}=\Gamma_{\textrm{R}}=\frac{\omega_{\textrm{S}}^{2}n_{\textrm{S}}^{2}g_{\textrm{R}}^{\textrm{B}}\hbar\omega_{\textrm{p}}\delta\omega_{\textrm{R}}}{3n_{\textrm{p}}\pi
c}\,.$ (10)

The cavity enhancement is given by a Lorentzian with amplitude
$F_{\textrm{P}}$ [39]. We approximate the Raman gain-profile with a normalized
Lorentzian [60]. We assume that the cavity is resonant with the Raman
scattered light $\omega_{\textrm{S}}=\omega_{\textrm{R}}$:

$\displaystyle\frac{1}{\tau_{\textrm{R}}^{\textrm{cav}}}=\frac{\omega_{\textrm{S}}^{2}n_{\textrm{S}}^{2}g_{\textrm{R}}^{\textrm{B}}\hbar\omega_{\textrm{p}}\delta\omega_{\textrm{R}}}{3n_{\textrm{p}}\pi
c}\int_{0}^{\infty}\textrm{d}\omega$
$\displaystyle\times\frac{2}{\pi}\frac{\delta\omega_{\textrm{R}}}{4(\omega-\omega_{\textrm{S}})^{2}+\delta\omega_{\textrm{R}}^{2}}$
$\displaystyle\times
F_{\textrm{P}}\frac{\delta\omega_{\textrm{S}}^{2}}{4(\omega-\omega_{\textrm{S}})^{2}+\delta\omega_{\textrm{S}}^{2}}\,.$
(11)

For $\omega_{\textrm{S}}\gg\delta\omega_{\textrm{S}}$,
$\delta\omega_{\textrm{R}}$ the integrand is close to 0 for $\omega=0$, so the
lower limit of the integral can be extended to negative infinity to obtain an
analytical solution:

$\frac{1}{\tau_{\textrm{R}}^{\textrm{cav}}}=\Gamma_{\textrm{R}}^{\textrm{cav}}=\frac{\omega_{\textrm{S}}^{2}n_{\textrm{S}}^{2}g_{\textrm{R}}^{\textrm{B}}\hbar\omega_{\textrm{p}}\delta\omega_{\textrm{R}}}{3n_{\textrm{p}}\pi
c}\frac{F_{\textrm{P}}\delta\omega_{\textrm{S}}}{\delta\omega_{\textrm{R}}+\delta\omega_{\textrm{S}}}\,.$
(12)

The Purcell enhancement of the system is given by:

$\frac{\Gamma_{\textrm{R}}^{\textrm{cav}}}{\Gamma_{\textrm{R}}}=F_{\textrm{P}}\frac{\delta\omega_{\textrm{S}}}{\delta\omega_{\textrm{R}}+\delta\omega_{\textrm{S}}}.$
(13)

This equation resembles the expression for Purcell enhancement of a two-level
emitter in a regime in which the linewidth of the cavity and a coupled emitter
are comparable [79, 80, 81, 82]:

$\frac{\Gamma_{\textrm{R}}^{\textrm{cav}}}{\Gamma_{\textrm{R}}}=\frac{3}{4\pi^{2}}\left(\frac{\lambda_{\textrm{S}}^{\textrm{cav}}}{n_{\textrm{S}}}\right)^{3}\frac{1}{V_{\textrm{R}}}\frac{Q_{\textrm{S}}Q_{\textrm{R}}}{Q_{\textrm{S}}+Q_{\textrm{R}}}\,.$
(14)

The lasing threshold power in the steady state can be calculated from:

$\Gamma_{\textrm{R}}^{\textrm{cav}}N_{\textrm{p}}(N_{\textrm{S}}+1)=\frac{N_{\textrm{S}}}{\tau_{\textrm{S}}}\,.$
(15)

Using Eq. 6 and Eq. 7,

$\kappa_{\textrm{in}}P_{\textrm{p}}=\frac{N_{\textrm{S}}}{\tau_{\textrm{S}}}\left(\frac{1}{\Gamma_{\textrm{R}}^{\textrm{cav}}(N_{\textrm{S}}+1)}\left(\frac{1}{\tau_{\textrm{p}}}+\frac{\gamma}{\tau_{\textrm{R}}}\right)+1\right)\,.$
(16)

Taking into account $N_{\textrm{S}}\gg 1$ and
$\frac{1}{\tau_{\textrm{p}}}\gg\frac{1}{\tau_{\textrm{R}}}$:

$\kappa_{\textrm{in}}P_{\textrm{p}}=\frac{1}{\tau_{\textrm{S}}\tau_{\textrm{p}}\Gamma_{\textrm{R}}^{\textrm{cav}}}\,.$
(17)

With $\kappa_{\textrm{in}}=\eta/(\hbar\omega_{\textrm{p}})$:

$\frac{\eta}{\hbar\omega_{\textrm{p}}}P_{\textrm{p}}=\frac{\omega_{\textrm{S}}\omega_{\textrm{p}}}{Q_{\textrm{S}}Q_{\textrm{p}}}\frac{n_{\textrm{S}}n_{\textrm{p}}V_{\textrm{R}}}{2\hbar\omega_{\textrm{p}}c^{2}g_{\textrm{R}}^{\textrm{B}}}\frac{Q_{\textrm{S}}+Q_{\textrm{R}}}{Q_{\textrm{S}}}\,.$
(18)

We obtain the result for the lasing threshold:

$P_{\textrm{p}}=\frac{1}{\eta}\frac{2n_{\textrm{S}}n_{\textrm{p}}\pi^{2}}{\lambda_{\textrm{S}}^{\textrm{cav}}\lambda_{\textrm{p}}^{\textrm{cav}}g_{\textrm{R}}^{\textrm{B}}}\frac{V_{\textrm{R}}(Q_{\textrm{S}}+Q_{\textrm{R}})}{Q_{\textrm{S}}^{2}Q_{\textrm{p}}}\,.$
(19)

Using the experimental values summarized in Table 1 yield
$P_{\textrm{th}}=187.5\,\textrm{mW}$ as stated in the main text.

Table 1: Summary of experimental parameters $\lambda_{\textrm{p}}^{\textrm{cav}}$ | $634.57\,\textrm{nm}$ |  | $\lambda_{\textrm{S}}^{\textrm{cav}}$ | $693.13\,\textrm{nm}$
---|---|---|---|---
$n_{\textrm{p}}$ | $2.4$ |  | $n_{\textrm{S}}$ | $2.4$
$Q_{\textrm{p}}$ | $297\,000\pm 500$ |  | $Q_{\textrm{S}}$ | $6\,650\pm 50$
$Q_{\textrm{R}}$ | $8\,960\pm 290$ |  | $g_{\textrm{R}}^{\textrm{B}}$ |  $\sim 40\,\textrm{cm/GW}$ [61]
$\eta$ | $0.45$ |  | $V_{\textrm{R}}$ | $109.85\,\upmu\textrm{m}^{3}$

###  .6 Future directions

We now turn to discuss some limiting factors and further possible improvements
to this experiment. The double resonance condition is satisfied for the
combination of $t_{\textrm{a}}$ and $t_{\textrm{d}}$ for which both pump and
Stokes modes are resonant simultaneously. With the current top mirror design
(depth of crater, $d=1.65\,\upmu\textrm{m}$) and diamond thickness
$t_{\textrm{d}}=754.58\,\textrm{nm}$, a relatively large air-gap of
$t_{\textrm{a}}=4173.98\,\textrm{nm}$ is required to meet this condition for
the range of $\lambda_{\textrm{pump}}$ available. The large air-gap results in
a large $V_{\textrm{R}}$, and consequently a large lasing threshold.
Establishing the double resonance condition for a shorter air-gap will reduce
$V_{\textrm{R}}$ and consequently $P_{\textrm{th}}$.

We simulate the cavity for a wide range of $t_{\textrm{a}}$ and
$t_{\textrm{d}}$ using
$\lambda_{\textrm{p}}^{\textrm{cav}}=633.52\,\textrm{nm}$ and
$\lambda_{\textrm{S}}^{\textrm{cav}}=691.10\,\textrm{nm}$ as before (Fig. 9
(a)). Reducing the diamond thickness to $t_{\textrm{d}}=721.82\,\textrm{nm}$
satisfies the double-resonance condition for
$t_{\textrm{a}}=1789.75\,\textrm{nm}$. For this air-gap, we calculate
$V_{\textrm{R}}=20.42\,\upmu\textrm{m}^{3}$.

Figure 9: (a) Simulated cavity mode structure for
$\lambda_{\textrm{p}}^{\textrm{cav}}=633.52\,\textrm{nm}$ (blue) and
$\lambda_{\textrm{S}}^{\textrm{cav}}=691.90\,\textrm{nm}$ (red) as a function
of $t_{\textrm{a}}$ and $t_{\textrm{a}}$. The double resonance condition is
satisfied for the pair of $t_{\textrm{a,d}}$ where the two respective modes
cross. The green circle highlights the combination
$t_{\textrm{a}}=4173.98\,\textrm{nm}$ and $t_{\textrm{d}}=754.58\,\textrm{nm}$
used in this experiment. The black dashed line represents the depth of the
crater. This depth sets a lower limit on the possible $t_{\textrm{a}}$ in the
current cavity geometry. (b) Simulated dependence of the $Q$-factor with air-
gap thickness for fixed
$\lambda_{\textrm{p}}^{\textrm{cav}}=633.52\,\textrm{nm}$ (blue) and
$\lambda_{\textrm{S}}^{\textrm{cav}}=691.90\,\textrm{nm}$ (red). The solid
lines represent the $Q$-factor in the absence of any losses, while the dashed
lines represent the $Q$-factor in the presence of surface scattering with
surface roughness $\sigma_{\textrm{q}}=0.3\,\textrm{nm}$. The drop in
$Q$-factors at large air-gap thicknesses is attributed to clipping losses at
the top mirror. The experimentally measured $Q$-factors are indicated by the
stars.

An additional benefit of reducing $t_{\textrm{a}}$ becomes apparent on
simulating the behavior of the $Q$-factor with increased cavity length (Fig. 9
(b)). The $Q$-factor and the cavity round-trip loss
$\mathcal{L}_{\textrm{cav}}$ are linked via $Q=\frac{4\pi
L_{\textrm{cav}}}{\lambda\mathcal{L}_{\textrm{cav}}}$, where
$L_{\textrm{cav}}=t_{\textrm{a}}+L_{\textrm{0}}$. Here the term
$L_{\textrm{0}}$ describes the diamond thickness and the field penetration
into the DBR mirror coatings [54, 36]. For short cavity lengths, the
$Q$-factor increases linearly with $t_{\textrm{a}}$. However, for large cavity
lengths, the extent of the intensity mode waist at the top mirror,
$w_{\textrm{I}}$, becomes larger than the spherical extent of the mirror,
leading to beam clipping and a subsequent drop in the $Q$-factor [37, 36, 83].
For a spherical mirror with diameter $D$, the clipping losses are calculated
according to
$\mathcal{L}_{\textrm{clip}}=e^{-\nicefrac{{D^{2}}}{{2w_{\textrm{I}}^{2}}}}$,
where the beam waist $w_{\textrm{I}}$ evolves according to [36]

$w_{\textrm{I}}=\sqrt{\frac{\lambda
R_{\textrm{cav}}}{\pi}}\times\left(\frac{R_{\textrm{cav}}}{(t_{\textrm{a}}+\frac{t_{\textrm{d}}}{n_{\textrm{d}}})}-1\right)^{-\frac{1}{4}}\,.$
(20)

In Fig. 9 (b), a drop in $Q$-factor is expected for ${t_{\textrm{a}}\gtrsim
3.5\,\upmu\textrm{m}}$, a consequence of clipping losses. Therefore, a shorter
$t_{\textrm{a}}$ will have the added benefit of preserving a high $Q$-factor.
Here, the value of $R_{\textrm{cav}}=11\,\upmu\textrm{m}$ and
$D=6\,\upmu\textrm{m}$ are extracted from a scanning confocal microscope
(Keyence Corporation).

Using this model for $t_{\textrm{d}}=721.82\,\textrm{nm}$ and
$t_{\textrm{a}}=1789.75\,\textrm{nm}$ (black circle in Fig. 9 (a)), we find a
theoretical $Q_{\textrm{p}}=406\,060$, $Q_{\textrm{S}}=13\,550$, $\eta=0.81$
and consequently $P_{\textrm{th}}=4.94\,\textrm{mW}$.

The diamond surface introduces scattering losses which should be taken into
account. Surface scattering can be incorporated in the transfer-matrix
simulations according to Ref. [84]. Motivated by typical roughness
measurements reported by Ref. [73, 36], including a scattering layer with
surface roughness ${\sigma_{\textrm{q}}=0.3\,\textrm{nm}}$, reduces the
$Q$-factor to $Q_{\textrm{p}}^{\textrm{scat}}=258\,780$ and
$Q_{\textrm{S}}^{\textrm{scat}}=13\,480$. Consequently, the additional loss-
channel reduces $\eta^{\textrm{scat}}=0.59$. The reduction in the $Q$-factor
increases the lasing threshold to
$P_{\textrm{th}}^{\textrm{scat}}=10.71\,\textrm{mW}$. Finally, increasing the
thickness of the diamond membrane provides a way to reduce further the lasing
threshold on the account of a larger $Q$-factors offered by the longer
effective cavity length. Applying the same method as in Fig. 9 (a), we find
that the double-resonance condition is established for
$t_{\textrm{d}}=3361\,\textrm{nm}$ and $t_{\textrm{a}}=1726\,\textrm{nm}$.
Using these values and simulating a loss-less cavity yields
$P_{\textrm{th}}=0.78\,\textrm{mW}$. Including surface scattering
($\sigma_{\textrm{q}}=0.3\,\textrm{nm}$) increases the threshold to
$P_{\textrm{th}}=1.87\,\textrm{mW}$.

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