# High-efficiency coupled-cavity optical frequency comb generation

M. P. Mrozowski  1,* J. Jeffers  1 and J. D. Pritchard1  1 Department of
Physics, University of Strathclyde, SUPA, Glasgow G4 0NG, UK
<EMAIL_ADDRESS>
###### Abstract

We present a high efficiency source of picosecond pulses derived from a dual
cavity optical frequency comb generator. This approach overcomes the
limitations of single cavity comb generators that are restricted to
efficiencies of a few percent. We achieve picosecond pulses with GHz
repetition rates offering over a hundred times higher output efficiency than a
single cavity design and demonstrate tuning of pulse width by varying the
modulation depth of the intra-cavity electro-optic modulator. These results
provide a wavelength-agnostic design with a compact footprint for the
development of portable picosecond pulsed laser systems for timing, metrology,
and LIDAR applications.

††journal: oe††articletype: Research Article

##  1 Introduction

Optical frequency comb technology underpins a wide range of applications in
sensing and metrology [1, 2, 3]. As well as enabling precision measurements,
frequency combs also provide an important resource for generating large
numbers of temporal and frequency modes for realising scalable quantum systems
[4]. Typically these experiments utilise combs derived from bulky, high power
and optically pumped mode-locked laser systems generating temporal pulses with
widths ranging from a few fs to 100s of ps, which come with a significant
price-tag ($\sim\pounds$100k) and a limited repetition rate ($\sim$ 1 - 100
MHz). These mode-locked laser systems additionally find applications in
quantum LIDAR via efficient generation of heralded single photons, for example
in a birefringent single mode optical fiber [5, 6].

One route towards compact, low power frequency combs is to utilise the large
optical non-linearity of whispering gallery mode microcomb based devices [7,
8], which can be fabricated on integrated photonic chips. These devices offer
very large free spectral ranges, but material losses in waveguide technologies
limit the wavelengths over which these devices can be utilised and they cannot
be dynamically tuned.

An alternative method is the optical frequency comb generator (OFCG) based on
an intra-cavity electro-optical modulator (EOM) which can convert a CW pump
into a mode-locked pulse train[9, 10, 11, 12]. These simple devices are able
to produce optical pulses with extremely high repetition rates up to 20 GHz,
whilst achieving pulse widths of order hundreds of ps down to a few hundred
femtoseconds[12]. Advantages of this approach are the design can be adapted to
operate at arbitrary pump wavelength using appropriate optics, the repetition
rate and pulse width can be dynamically adjusted by changing the EOM
modulation depth, as well as a very low timing jitter that makes it suitable
for the distribution of precision timing signals [13]. As an example of OFCG
verstability, it can be combined with spectral pulse shaping to realise a ps-
pulse arbitrary waveform generator in the telecom band [14]. The main drawback
of the single cavity OFCG is the intrinsically low output efficiency
(typically $<1\%$) which arises from the imperfect impedance matching when
coupling light into the OFCG [9].

In this paper we present a low cost and compact source of few-ps optical
pulses based on a dual-cavity OFCG[15] that overcomes the efficiency
limitation of the single cavity design. We demonstrate an output efficiency
exceeding 80%, achieving a minimum pulse width of 1.77 ps, in excellent
agreement with a parameter-free model of the coupled cavity system. This
provides a simple, low-power source suitable for applications in quantum LIDAR
[16].

##  2 Coupled Cavity OFCG

Figure 1:  Dual-cavity OFCG formed from a CW seed laser injected into an
impedance matched input cavity (M1 & M2) with free spectral range
$\mathrm{FSR_{in}}$ to pump the EOM cavity (M2 & M3) with free spectral range
$\mathrm{FSR_{EOM}}$ chosen to be an integer multiple of $=f_{m}$, the EOM
modulation frequency. This configuration generates a high efficiency optical
frequency comb, with short temporal pulses at a repetition rate of $2f_{m}$
and a broad frequency spectrum showing a sharp cut-off at $\pm\nu_{c}$, the
characteristic mode resonant with the input cavity.

A single cavity OFCG source uses an EOM inside an optical cavity driven
resonantly at the cavity free spectral range (FSR) to generate a comb of
phase-coherent sidebands that convert a CW pump laser into a train of short
optical pulses that have a Lorentzian line shape and a repetition rate equal
to twice the modulation frequency [17, 11, 10]. The output efficiency of the
single cavity OFCG is [13]

$\eta_{\mathrm{SC}}=\frac{(1-\mathrm{R})^{2}}{\pi\mathrm{R}(1-\mathrm{R}\eta_{c})\beta},$
(1)

where $\mathrm{R}$ is the power reflectivity of the resonant cavity mirrors,
$\eta_{c}$ is the crystal transmission efficiency and $\beta$ is the
modulation depth in radians. The pulse width at FWHM intensity is given by
[17]

$\Delta\tau_{p}=\frac{1}{2F\beta f_{m}},$ (2)

where $F$ is the finesse of the resonant cavity and $f_{m}$ is the modulation
frequency of the EOM. Equations (1) and (2) show that it is not possible to
generate short optical pulses with high output efficiency, as increasing
either cavity finesse or modulation depth increases the loss from the central
cavity mode, resulting in an impedance mismatch that inhibits efficient input
coupling into the cavity and hence low output, with <1$\%$ being typical for
experiments [11, 12].

This limitation can be overcome using a coupled cavity configuration, as first
demonstrated in [15] and shown schematically in Fig. 1. Here the system
consists of three mirrors M1-3, with power reflectivities $\mathrm{R_{1-3}}$
forming two optical cavities, the input cavity (M1-M2) and the EOM cavity
(M2-M3) containing the modulator crystal. In this configuration, the presence
of the input cavity with $\mathrm{R}_{1}=\mathrm{R}_{2}$ allows for critical
impedance matching of the OFCG system, drastically increasing the output power
and theoretically enabling 100% input coupling into the EOM cavity formed by
M2-M3.

In our setup we drive the EOM crystal at a modulation frequency $f_{m}$ chosen
to be an integer multiple $n$ of $\mathrm{FSR_{EOM}}$, the FSR of the EOM
cavity. For the coupled cavity setup, an additional restriction on the
frequency width of the generated comb arises when the $n^{\mathrm{th}}$
sideband of the EOM becomes resonant with the $j^{\mathrm{th}}$ mode of the
input cavity

$n\times f_{m}=j\times\mathrm{FSR_{in}}.$ (3)

To maximise the comb-width (and hence minimise pulse width),
$\mathrm{FSR_{in}}$ is chosen to have a large common denominator with $f_{m}$,
and a large finesse to reduce the input cavity linewidth. The first sideband
that is completely resonant with the input cavity is called the characteristic
mode $\nu_{c}$, which defines the maximum sideband order populated in the
comb.

The added complexity of the dual-cavity operation and characteristic mode
causes the efficiency and pulse-width to deviate from the simple formulae
presented above for the single cavity case. Instead it is necessary to solve
numerically the detailed model presented in Ref. [16] describing the interplay
of the two cavities. Experimental parameters were chosen by using this model
to find the required mirror reflectivities and cavity FSRs to give high
efficiency pulses with a target duration of 1.7 ps and at a 4.8 GHz repetition
rate. In the figures below we also use this model for theoretical comparison
to experimental data. Furthermore, it is essential to ensure that the mirror
radii of curvature are chosen to give a matching waist for the fundamental
modes of both cavities in order to maintain spatial mode-matching conditions
and suppress excitation of transverse modes of the EOM cavity.

##  3 Experiment Setup

Figure 2:  Schematic of the experimental setup showing a CW laser diode
($\mathrm{LD}$) with a tandem isolator locked to the input cavity using a
Pound-Drever-Hall lock. The EOM cavity length is stabilised using a piezo
actuator on M3 locked using an error signal derived from demodulation of a
fast-photodiode signal. Frequency and time domain of the output pulses are
analysed in a : A Fabry-Perot analysis cavity. b: A auto-correlation setup
with time variable delay, here CC is a corner cube, TS is a motorised
translation stage used to generate time delay $\mathrm{2\Delta d}$/c and GaP
is a slow GaP photodetector.

The experimental setup for our realisation of the dual-cavity OFCG is shown
schematically in Fig. 2. The seed light for the system was provided by a
distributed bragg reflector (DBR) laser diode at 780 nm, with the maximal
power output to the experiment of 30 mW. Due to strong optical feedback from
the input cavity a 60 dB tandem isolator is used, and the diode laser is fiber
coupled into a single mode polarisation maintaining fiber to allow efficient
spatial mode-matching into the input cavity.

The input cavity (M1-M2) is a plano-concave cavity with $R_{1}=R_{2}=0.99$
formed using an input radius of curvature of 1000 mm and a separation of 12.6
mm tuned to obtain a measured FSR of $\mathrm{FSR_{in}}$ = 11.85 $\pm$ 0.05
GHz. The cavity is constructed using a monolithic brass mount to minimise
sensitivity to vibration. Mirror M2 is oriented with the antireflection (AR)
coated surface located inside the input cavity to supress losses within the
EOM cavity, resulting in a measured input cavity finesse of $280\pm 10$. The
seed light is stabilised to the input cavity using a Pound-Drever-Hall lock,
using current modulation to add sidebands to the laser output at 4 MHz. We
obtain 95% transmission of light into the EOM cavity when the laser is locked
to the input cavity limited by the spatial mode-matching of light into the
cavity.

The intracavity EOM is chosen to have a resonant frequency of $f_{m}=2.39$
GHz, with modulation depth $\beta$ in the range of 0.3 to 2.05 rad used in
experiments below by tuning the applied microwave power. The crystal inside a
free space EOM has the following parameters, crystal length
$\mathrm{L_{c}}=27$ mm, group velocity dispersion GVD = 447 fs2/mm and
transmission efficiency $\eta_{c}$ = 0.996.

Figure 3: Output OFCG efficiency for the single cavity OFCG -
$\mathrm{\eta_{SC}}$ and the coupled cavity OFCG - $\mathrm{\eta_{CC}}$. The
black square data-point is efficiency recorded in [15]. Figure 4: a: Frequency
spectrum seen on the analysis cavity. Additional modes seen on the plot can be
attributed to the imperfect transverse mode suppression in the analysis
cavity, and to frequency combs overlapping due to $\mathrm{FSR_{a}}$ being
narrower than the produced frequency comb. b: The normalised field
autocorrelation signal, the fit function used to determine the FWHM pulse
width $\Delta\tau_{p}$ = 1.77 $\pm$ 0.04 ps, highlighted on the plot with the
red colour. The inset plot shows the field autocorrelation signal on a
femtosecond scale. Both traces were obtained for $\beta$ = 2.05 rad.

The EOM cavity is formed by adding mirror M3 with reflectivity $R_{3}=0.96$
and radius of curvature of 300 mm, chosen to mode-match the input cavity waist
of 150 $\mu$m. The finesse of the EOM cavity is restricted to 100 due to the
losses caused by the finite crystal transmission and the imperfect AR coatings
of the crystal facet. Following initial alignment, the EOM cavity length is
adjusted to obtain a free spectral range $\mathrm{FSR_{EOM}}$ = 478 MHz,
corresponding to the fifth subharmonic of the modulation frequency $f_{m}$. In
addition to matching the FSR by adjusting cavity length, the position of the
EOM crystal must also be adjusted to ensure the crystal is located at an anti-
node of the microwave modulation frequency to ensure the relative phase of
successive cavity round-trips add constructively. The EOM crystal is initially
coarsely placed at $\sim\lambda_{m}$ from M2, and then its location is
adjusted using a translation stage to maximise the suppression of the
fundamental mode to optimise the round-trip RF phase matching condition [18].
Here we use the fact that the power in the central frequency decreases as the
modulation depth increases. The EOM cavity is stabilised by using a piezo
actuator on M3. A dispersive error signal is obtained by picking off a
fraction of the light transmitted through M3 and directing it to a fast
photodiode. The fast photodiode signal is then demodulated at the modulation
frequency $f_{m}$ and low-pass filtered. We utilise the fact that at the exact
resonance the power of the $f_{m}$ harmonic almost vanishes [12]. The error
signal is then amplified and a fast Red-Pitaya based PID controller[19] is
used to apply feedback to the piezo.

##  4 Experimental Results

To evaluate the performance of the coupled cavity OFCG, we compare the output
efficiency as a function of modulation depth against that obtained in the
single cavity configuration with mirror M1 removed. For the single cavity data
the OFCG cavity is locked as described above with the input laser free-
running. Results are shown in Fig. 3, with errorbars representing the standard
deviation of 30 measurements at each modulation depth. The maximum output
efficiency for the coupled cavity configuration is $81\pm 2$% at
$\beta=1~{}$rad, more than a 100$\times$ improvement compared to the single
cavity configuration, and $7\times$ higher than the previous demonstration of
a coupled cavity setup [15].

The single cavity OFCG efficiency data $\eta_{\mathrm{SC}}$ are compared to
parameter free theoretical curve taken from Eq. (1), whilst the theoretical
coupled cavity efficiency $\eta_{\mathrm{CC}}$ was generated using the
numerical model from Ref. [16], using parameters given in Sec. 3. We obtain
good agreement with theory for the high values of $\beta$ > 1.25 but a slight
deviation for $\beta<0.75$. This can be attributed to the significantly lower
bandwidth of the error signal when $\beta$ is reduced in both single and
coupled cavity OFCG configurations. In the case of coupled cavity system,
additional errors occur at low modulation depth due to the increased leakage
of fundamental mode seed light leaking from the EOM cavity into the input
cavity which perturbs it’s lock, preventing stable operation at lower values
of $\beta$.

To characterise the pulse train output from the dual cavity OFCG, some of the
output light is picked off and directed to an analysis cavity and an
autocorrelation module in order to characterise the frequency and temporal
properties as shown in Fig. 3. The analysis cavity consists of two mirrors
with identical power reflectivity $\mathrm{R_{a}}$ = 0.99 and an FSR of
$\mathrm{FSR_{a}}$ = 130 GHz. The autocorrelation module is a scanning
Michelson interferometer using travelling corner cubes to adjust relative
pulse delay between the two arms [20]. The interferometer output is detected
using a slow GaP photodiode and a high-gain, low noise current amplifier.

The frequency spectrum for the dual-cavity OFCG operating at $\beta=2.05$ rad
is shown in Fig. 4a, corresponding to a FWHM bandwidth of $\sim$120 GHz. Using
the fact the OFCG output pulses should be Lorentzian, this corresponds to a
transform-limited FWHM pulse width of $\sim$1.75 ps. Here the spectral
envelope shows a deviation from the monotonic exponential decay with sideband
order expected for a Lorentzian, however calculating the Fourier transform of
the observed spectrum with the appropriate phase relationship caused by the
round-trip phase accumulated in the EOM cavity shows excellent agreement with
a Lorentzian pulse shape. This deviation in spectral output is likely due to
residual amplitude modulation, which has been observed in previous
experiments[17, 21, 22]. Comparing the single and dual cavity output spectra
we observe an identical envelope verifying this arises from the EOM cavity.

To measure the pulse width independently we record the field autocorrelation
as shown in Fig. 4b, using the observed interference fringes to calibrate
pulse delay. Due to the relatively low peak pulse power we are unable to
observe a two-photon absorption feature using the GaP detector, and this data
therefore corresponds to the linear field autocorrelation signal [20]. The
temporal envelope is fit using a Lorentzian to obtain a FWHM pulse width to be
$1.77\pm 0.04$ ps in good agreement with the spectral characterisation using
the analysis cavity.

Figure 5: The FWHM pulse width as a function of the EOM cavity frequency
detuning. Data presented here was recorded for modulation depth $\beta$ = 2.05
rad.

The OFCG output pulse width depends strongly on the characteristic mode
$\nu_{c}$ which becomes resonant with the input cavity. The monolithic design
of the input cavity prevents scanning $\mathrm{FSR_{in}}$. Instead we scan the
FSR of the EOM cavity by temperature tuning the EOM resonance frequency and
using the lock feedback to compensate the change by adjusting the EOM cavity
length. Results are shown in Fig. 5, there the effect of changing the ratio
between $\mathrm{FSR_{in}}$ and $f_{m}$ on the measured pulse width can be
observed and again we obtain good agreement with our theoretical model using
$f_{m}$ as a variable parameter.

Finally, we measure pulse width and hence extract peak power as a function of
the modulation depth. Results are shown in Fig. 6, with the drop in output
efficiency for $\beta>1$ observed in Fig. 4 compensated by the reduction in
pulse width $\propto 1/\beta$ corresponding to a peak pulse power of 2.6 W.
Again we see good agreement with the theoretical model of Ref. [16], with a
larger deviation in the predicted peak power at lower modulation depths
primarily due to the reduction in input cavity lock stability. These results
imply higher power would be possible using increased modulation depth, however
for $\beta>\pi$ the output pulse spacing becomes unstable.

Figure 6: The FWHM pulse width and peak power per pulse as a function of the
modulation depth $\beta$. $P_{\mathrm{peak}}$ was recorded for input optical
power equal to 30 mW.

##  5 Conclusion

In conclusion, we present a highly efficient, high repetition rate source of
pulsed light. With an input power of 30 mW, peak powers of $\sim$ 2.6 W were
observed at a repetition rate of 4.78 GHz, achieving pulse widths down to 1.77
ps limited by the maximum accessible modulation depth of the EOM. Our results
show excellent agreement with a parameter free theoretical model for the OFCG
performance [16], and demonstrate the ability to realise an OFCG with over 80%
output efficiency. To obtain higher output power, we have repeated the
characterisation using a high power, narrow linewidth Ti:Sapph laser enabling
use of up to 300 mW into the OFCG. Our measurements show the OFCG retains the
same high efficiency and pulse width observed with low power laser, however
even for this increased peak power of up to 26 W we still are unable to
observe a two-photon intensity autocorrelation spectrum using GaP detector.
The current limitation in operating with higher input powers arises from the
damage threshold of the AR coating on the EOM of 20 W/mm2. This, combined with
the relatively small active crystal aperture of $2\times 2$ mm required to
maintain a GHz resonance frequency limits us to 300 mW seed. In future, this
can be overcome using alternative coatings or EOM crystal materials, or for
example operating at telecom wavelengths to benefit from higher damage
thresholds. Already the output pulse power demonstrated here with the dual
cavity OFCG is sufficient to enable heralded photon pair generation in a
birefringent optical fiber [16], and this design paves the way to realising
compact, low cost and high repetition rate picosecond pulses for a wide range
of potential applications in metrology, sensing and quantum lidar.

Acknowledgements We thanks E. Riis for useful discussions and loan of
equipment. This project is funded by the UK Ministry of Defence. Data
availability The datasets used in this work are available at [23].

## References

  * [1] T. Udem, J. Reichert, R. Holzwarth, and T. W. Hänsch, “Accurate measurement of large optical frequency differences with a mode-locked laser,” Opt. Lett. 24, 881–883 (1999).
  * [2] H. R. Telle and U. Sterr, _Generation and Metrological Application of Optical Frequency Combs_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 2001), pp. 295–313.
  * [3] T. Fortier and E. Baumann, “20 years of developments in optical frequency comb technology and applications,” Communications Physics 2, 153 (2019).
  * [4] M. Kues, C. Reimer, J. M. Lukens, W. J. Munro, A. M. Weiner, D. J. Moss, and R. Morandotti, “Quantum optical microcombs,” Nature Photonics 13, 170–179 (2019).
  * [5] B. J. Smith, P. Mahou, O. Cohen, J. S. Lundeen, and I. A. Walmsley, “Photon pair generation in birefringent optical fibers,” Opt. Express 17, 23589–23602 (2009).
  * [6] D. G. England, B. Balaji, and B. J. Sussman, “Quantum-enhanced standoff detection using correlated photon pairs,” Phys. Rev. A 99 (2019).
  * [7] A. Pasquazi, M. Peccianti, L. Razzari, D. J. Moss, S. Coen, M. Erkintalo, Y. K. Chembo, T. Hansson, S. Wabnitz, P. Del’Haye, X. Xue, A. M. Weiner, and R. Morandotti, “Micro-combs: A novel generation of optical sources,” Physics Reports 729, 1–81 (2018). Micro-combs: A novel generation of optical sources.
  * [8] T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. L. Gorodetsky, “Dissipative kerr solitons in optical microresonators,” Science 361, eaan8083 (2018).
  * [9] T. Kobayashi, T. Sueta, Y. Cho, and Y. Matsuo, “High-repetition-rate optical pulse generator using a fabry-perot electro-optic modulator,” Applied Physics Letters 21, 341–343 (1972).
  * [10] M. Kourogi, K. Nakagawa, and M. Ohtsu, “Wide-span optical frequency comb generator for accurate optical frequency difference measurement,” IEEE J. Quantum Electron. 29, 2693–2701 (1993).
  * [11] R. P. Kovacich, U. Sterr, and H. R. Telle, “Short-pulse properties of optical frequency comb generators,” Appl. Opt. 39, 4372–4376 (2000).
  * [12] S. Xiao, L. Hollberg, and S. A. Diddams, “Generation of a 20 ghz train of subpicosecond pulses with a stabilized optical-frequency-comb generator,” Opt. Lett. 34, 85–87 (2009).
  * [13] S. Xiao, L. Hollberg, N. R. Newbury, and S. A. Diddams, “Toward a low-jitter 10 ghz pulsed source with an optical frequency comb generator,” Opt. Express 16, 8498–8508 (2008).
  * [14] Z. Jiang, D. E. Leaird, C.-B. Huang, H. Miao, M. Kourogi, K. Imai, and A. M. Weiner, “Spectral line-by-line pulse shaping on an optical frequency comb generator,” IEEE Journal of Quantum Electronics 43, 1163–1174 (2007).
  * [15] G. M. Macfarlane, A. S. Bell, E. Riis, and A. I. Ferguson, “Optical comb generator as an efficient short-pulse source,” Opt. Lett. 21, 534–536 (1996).
  * [16] M. P. Mrozowski, J. Jeffers, and J. D. Pritchard, “A practical compact source of heralded single photons for a simple detection LIDAR,” in _Quantum Communications and Quantum Imaging XIX,_ vol. 11835 K. S. Deacon and R. E. Meyers, eds., International Society for Optics and Photonics (SPIE, 2021), p. 1183508.
  * [17] T. Kobayashi, T. Sueta, Y. Cho, and Y. Matsuo, “High-repetition-rate optical pulse generator using a fabry-perot electro-optic modulator,” Appl. Phys. Lett. 21, 341–343 (1972).
  * [18] A. S. Bell, G. M. Mcfarlane, E. Riis, and A. I. Ferguson, “Efficient optical frequency-comb generator,” Opt. Lett. 20, 1435–1437 (1995).
  * [19] M. A. Luda, M. Drechsler, C. T. Schmiegelow, and J. Codnia, “Compact embedded device for lock-in measurements and experiment active control,” Review of Scientific Instruments 90, 023106 (2019).
  * [20] V. Wong and I. A. Walmsley, “Analysis of ultrashort pulse-shape measurement using linear interferometers,” Opt. Lett. 19, 287–289 (1994).
  * [21] L. R. Brothers and N. C. Wong, “Dispersion compensation for terahertz optical frequency comb generation,” Opt. Lett. 22, 1015–1017 (1997).
  * [22] M. Kourogi, B. Widiyatomoko, Y. Takeuchi, and M. Ohtsu, “Limit of optical-frequency comb generation due to material dispersion,” IEEE Journal of Quantum Electronics 31, 2120–2126 (1995).
  * [23] M. P. Mrozowski, J. Jeffers, and J. D. Pritchard. DOI to be added in proof.