Residual phase noise suppression for Pound-Drever-Hall cavity stabilization with an electro-optic modulator

Mamoru Endo; Thomas R. Schibli

doi:10.1364/OSAC.1.000116

1. Introduction

Narrow linewidth lasers locked to ultra-stable optical reference cavities have become invaluable tools for precision spectroscopy and metrological applications, including gravitational wave sensing [1], very-long baseline interferometry [2,3], ultra-low noise photonic microwave generation [4–6], optical clocks [7,8], and many more. Recent progress with ultra-stable reference cavities enables mHz-linewidth lasers, e.g. with a cryogenically cooled, single-crystal silicon cavity for the next-generation optical clocks [9]. Such lasers are reaching frequency instabilities in the range of $10^{- 16}$ and below at 1 s gate time. For clock applications, the stability at long timescales (that is, phase noise at very low Fourier frequencies) is of particular interest. For photonic microwave generation, however, noise at short timescales (milliseconds to microseconds) is much more critical [10]. For instance, to produce an ultra-low noise microwave carrier at 10 GHz via optical frequency division with a noise floor below −180 dBc/Hz, an optical phase noise of ~-100 dBc/Hz is required at a wavelength of 1550 nm. This corresponds to an optical frequency noise power spectral density (PSD) of $2 \cdot 10^{- 2} {Hz}^{2} /Hz$ at 10 kHz offset. Even commercial ultra-narrow cw lasers (often advertised as ‘sub-Hz’ lasers) show free-running frequency noise densities 2 to 4 orders of magnitude above that requirement. The challenge of these poses is that the suppression of this intrinsic laser noise will require ~20 - 40 dB gain at 10 kHz offset. This demands large feedback bandwidths to suppress the phase noise at the low frequency range. As a rule of thumb, a loop filter that provides 40 dB gain at 10 kHz needs to have a stable unity gain at ~1 MHz. This requires a feedback loop (including the actuators) with low phase lag.

The Pound-Drever-Hall (PDH) method is widely used for stabilizing a laser’s longitudinal mode to an optical reference cavity [11,12]. The laser is locked to the cavity by feeding back to its optical frequency via various methods. A piezo actuator (PZT) in the laser cavity can control the optical frequency, but typical PZTs have strong mechanical resonances between 10 kHz to 100 kHz. These resonances limit the achievable feedback bandwidth to typically a few tens of kilohertz. To suppress these resonances, a specially designed PZT and PZT mount are required [13,14], but most commercially available cw sources lack such techniques. In the case of laser diodes, pump power modulation can also be used for tuning the laser’s frequency. The achievable bandwidth in that case is typically much wider than that of PZTs, but the frequency to current response typically contains a sign reversal around a few 100 kHz. This is due to a competition between thermal and electrical carrier-induced effects, which have the opposite effect on the frequency tuning of the laser. Current modulation may also increase the intensity noise of the laser output. For this reason, and the fact that off-the-shelf cw lasers can typically not be upgraded with faster intra-cavity modulators, extra-cavity acousto-optic modulators (AOMs) are commonly used to improve the achievable loop bandwidth [15,16]. An AOM shifts an optical frequency by its modulation frequency, so that the optical frequency can be controlled by tuning the modulation frequency via a voltage-controlled oscillator. The feedback bandwidth of the AOM is usually limited by the first-order phase delay caused by the speed of sound in the AO crystal. This phase delay typically limits the maximum useable feedback bandwidth to a few hundred kHz. Due to the frequency-dependent Brag angle in AOMs, the AOM is often used in a double-pass configuration. Due to the finite diffraction efficiency, especially at NIR wavelengths, this configuration often results in notable optical losses and an increased complexity. The need for a high-power RF driver and the associated heat in the optics setup can be an additional concern in high precision stabilization schemes.

An alternative method is based on an optical frequency shifting with the use of an electro optic modulators (EOMs). Frequency shifters with serrodyne modulator, which employ a sawtooth phase modulation to generate a frequency shift of the optical signal, or dual-parallel EOM (DPEOM) based frequency shifter, have been demonstrated by several groups [17–19]. The advantages of these methods are wide tuning range (up to several GHz), fast modulation bandwidth (multi-MHz) and lower loss than the AOM approach configuration. Especially, R. Kohlhaas et al. [18] apply serrodyne modulation method to the PDH cavity stabilization, in which a single EOM is used to generate the PDH phase side bands and the frequency shift signals. This simplifies the optical setup, when compared to the AOM approach. However, frequency shifting methods mentioned above cannot perfectly suppress the original carrier component because of the imperfect modulation. The serrodyne modulation method requires an ideal sawtooth signal which contains infinitely high frequency components, and the DPEOM method needs a precision bias voltage control for the DPEOM. Another EOM-based approach was demonstrated by M. Musha et al. [20], where a second EOM was used as a fast actuator, while slow frequency drifts were controlled by a PZT inside the laser cavity. This method does not create the unwanted sidebands caused by the imperfections in the modulation used by frequency shifting methods.

Here, we modified M. Musha’s apparatus [20] with the use of a single EOM to simplify the PDH cavity stabilization. The experimental apparatus is significantly simplified, only an electronic circuit, comprised of a loop filter, is required. Since this approach does not require additional optical elements, such as a second EOM or an AOM, no additional optical losses incur. Despite the simplicity of the setup, this method allows further noise suppression by an additional ~30 dB at 10 kHz offset and more at lower frequencies. By adding this additional feedback loop, one can enhance the feedback bandwidth to multi-MHz, independent of the actuator speed in the laser itself. This method could also be used for stabilizing a laser to non-traditional reference cavities, such as whispering-gallery-mode resonators [21] or optical delay lines.

2. Method

The instantaneous carrier frequency $ν (t)$ and the instantaneous phase $ϕ (t)$ of an optical carrier are related by:

ν (t) = ν_{0} + Δ ν (t) = ν_{0} + \frac{1}{2 π} \frac{d ϕ (t)}{d t},

where

ν_{0}

is the initial carrier frequency at

t = 0

, and

Δ ν (t)

is the frequency deviation from

ν_{0}

. That is, the frequency shift can be obtained by changing the carrier phase. However, for achieving a constant frequency shift, an endless phase shift is required. Since the maximum achievable phase shift of an EOM is limited, the serrodyne modulation technique [17,18] or a ramp function [22] can be employed, with the aforementioned limitations. In the following, we assume that the carrier frequency is roughly locked to the target frequency, e.g. via a low bandwidth (~15 kHz) feedback loop. This way we can focus on eliminating the residual frequency fluctuations, that is, we only focus on the AC components from Eq. (1).

Let us now consider the AC component of the phase after passing the carrier through an EOM:

ϕ (t) = \frac{π V (t)}{V_{π}},

where

V_{π}

is the half wave voltage of the EOM (a few Volts for waveguide EOMs and ~100 V or higher for bulk EOMs) and

V (t)

is the applied voltage to the EOM. The frequency modulation caused by the EOM can then be written as

Δ ν (t) = \frac{1}{2 π} \frac{d V (t)}{d t} .

To compensate for the residual AC fluctuations

δ \sin (2 π f t)

of the carrier, where

δ

is the frequency shift at the Fourier frequency

f

, the required voltage of the EOM is the root of the differential equation

- \frac{1}{2 V_{π}} \frac{d V (t)}{d t} = δ \sin (2 π f t) .

The relevant solution can be written as

V (t) = \frac{δ}{π f} V_{π} \cos (2 π f t) .

From Eq. (5), to achieve the frequency shift

δ

at the Fourier frequency of

f

, the sinusoidal signal with the peak-to-peak voltage of

V_{pp} = \frac{2 δ V_{π}}{π f},

and the frequency of

f

is required. For example, when

V_{π} = 4.2 V, δ = 5 kHz

, and

f = 10 kHz

(assuming typical condition for laser stabilization to the high-finesse cavity, with the waveguide EOM), the required

V_{pp}

is 13.4 V. This corresponds to the power of + 26.5 dBm at a 50-ohm load resistance. The required

V_{pp}

decreases at the higher offset frequencies proportional of the inverse of

f

. For a given experimental situation (i.e. an EOM with

V_{π}

, and the maximum peak-to-peak voltage of EOM driver of

V_{pp}

), the proposed method can eliminate a peak amplitude frequency fluctuation of

δ = \frac{π V_{pp}}{2 V_{π}} f,

at the Fourier frequency

f

. Note that, the actual noise suppression ratio is also dependent on the electronics, such as the loop filter’s bandwidth and delay or the phase lag and frequency limit of the RF filters used. Applying this to the PDH method, fluctuations at low offset frequencies (that is, slow fluctuations) are typically well suppressed by the slow feedback going directly to the tuning port of the laser. The main difficulty to achieve a tight lock is caused by the higher frequency components (e.g. higher than ~10 kHz). Based on above discussion, it can be shown that a commercial waveguide EOM usually has a sufficient dynamic range to eliminate the residual noise components. In other words, one could convert the residual frequency fluctuations after locking the laser to the cavity with the slow feedback loop, into phase noise. The residual root mean square (RMS) phase noise must not exceed the achievable RMS modulation depth of the EOM.

To use a single EOM for frequency noise suppression and PDH side-band generation, a frequency duplexer can be used to combine the few tens of megahertz PDH modulation signal with the DC – 10 MHz feedback signal. Sharing one EOM not only decreases the experimental complexity and cost, but also does not incur additional optical losses compared to the popular AOM approach. The major drawback of sharing a single EOM is that the output light has the PDH sidebands around the carrier. In situations, where these sidebands would pose an issue, a second EOM followed by a beam splitter could be used in front of the PDH EOM as demonstrated in [20]. This way, a side-band free output can be achieved. However, a carefully chosen PDH modulation frequency can typically avoid issues with said sidebands and a single EOM setup suffices. Moreover, in photonic microwave generations, tens of megahertz sidebands are typically not an issue, as the feedback bandwidth of the optical frequency divider is typically limited to less than 1 MHz.

3. Experiment and results

Figure 1(a)

Fig. 1 (a) Schematic apparatus. (b) (Top) The error signal obtained by the PDH method. (Bottom) The transmission signal obtained on D2. (c) The expanded trace of the top of (b). The peak-to-peak voltage difference between (b) and (c) is caused by the sampling rate difference. The asymmetric shape of (c) is due to the laser’s free-running jitter.

Download Full Size | PDF

shows a schematic of the apparatus for stabilizing a commercial, distributed-feedback erbium-doped fiber laser (‘DFB EDFL’, Koheras BASIK MIKRO E15 PM, NKT Photonics) to a high-finesse cavity with a free-spectral range of 3.08 GHz and a finesse of 400,000 at the wavelength of 1550 nm (‘High-F cavity’, Stable Laser Systems) placed in a temperature-controlled vacuum chamber. The DFB EDFL provides a single longitudinal mode with a wavelength of 1550.12 nm and an average power of 40 mW. The output wavelength can be controlled via temperature control and a PZT actuator for slow and fast feedback, respectively. The PZT for the fast feedback suffers from high-Q mechanical resonances around 20 kHz and 40 kHz and higher. These resonances limit the useable feedback bandwidth to below ~15 kHz. The laser output was connected to a fiber-coupled waveguide EOM (PM-0K5-10PFA-PFA, EOspace) with a 3-dB bandwidth of 10 GHz and an insertion loss of less than 2 dB. A direct-digital synthesizer (‘DDS’, DG4162, RIGOL) was used to generate the modulation signal for the PDH method at a frequency of 15 MHz. This signal was applied to the EOM through the high frequency port (‘HF’) of a duplexer (‘DP’), which had a cutoff frequency of 7 MHz. After the EOM, the light was guided through a half-wave plate (‘HWP’), a polarization beam splitter (‘PBS’) and a quarter-wave plate (‘QWP’) into the High-F cavity. The HWP could be used to adjust the input power to the cavity. The light reflected on the PBS in the forward direction was used as the stabilized output, and the light reflected in the backwards direction was used to generate the PDH error signal. The average power into the cavity was limited to 25 μW to avoid heating of the mirror coatings. The remaining light (~26 mW) was available as the stabilized output. The coupling efficiency to the cavity was approximately 60%. The light transmitted through the cavity was detected by a biased photodiode (‘D2’, G12180-003, Hamamatsu) for monitoring purposes. The light reflected from the cavity was used to produce the PDH error signal via a diode (‘D1’, G11193-02, Hamamatsu) that was connected to a bootstrapped cascode transimpedance amplifier [23]. This purpose-built amplifier enables shot-noise limited detection at of ~20–30 μA at a modest transimpedance gain of 20 kΩ and relatively high bandwidths (~40 MHz). The signal was then further amplified by a low noise amplifier (‘Amp’, MAN-1LN, Mini-Circuits) and the PDH error signal was generated by demodulation via a high-voltage output, balanced phase detector (‘PD’, MPD-1, Mini-Circuits) and an adjustable phase shifter ‘PS’.

The measured error and the transmission signals are shown in Fig. 1(b), 1(c). The slope of the error signal was calculated to 30.4 μV/Hz from Fig. 1(c). This value was used for calculating the residual frequency and phase-noise PSDs. Note that, the free-running laser linewidth (at most 100 Hz with an integration time of 120 μs) is narrower than that of the cavity (7.7 kHz), and the frequency scanning speed was approximately 300 MHz/s to acquire the data in Fig. 1(b) and 1(c). That is, during the cavity lifetime ( $1 / (2 π \times 7.7 kHz) ~ 21 μ s$ ), the laser frequency was scanned by ~6.2 kHz, which is less than the cavity linewidth. That supports the measured error signal slope. The initial laser stabilization was achieved via a proportional-integral (PI) loop filter (‘LP1’), a PZT driver (‘PZTD’, MDT694B, Thorlabs), and a PZT in the laser cavity of DFB EDFL as indicated by the blue shaded part in in Fig. 1(a). This loop is labeled as ‘slow FB’ due to the limited usable PZT bandwidth. As discussed above, the feedback bandwidth can be greatly increased by the addition of a second feedback loop to the EOM. This fast feedback loop consisted of a PI loop filter (‘LP2’), an EOM driver (‘EOMD’) and the shared EOM as shown in the red shaded region of Fig. 1(a). This loop is labeled as ‘fast FB.’ The maximum output peak-to-peak voltage of the EOMD is 20 V, which is sufficient to suppress the residual noise for our experiment (see Method section above). The feedback signal was applied to the EOM through the low frequency port (‘LF’) of the DP. The error signal was measured by an oscilloscope [not shown in the Fig. 1(a)] and a signal analyzer (PXA N9030A, Keysight). To eliminate the 15-MHz modulation signal, a low-pass filter (LPF) was applied before the signal analyzer. This LPF ultimately limits the usable loop bandwidth of the fast loop. A series of notch filters instead of the LPF could further increase the loop bandwidth of the ‘fast FB.’

Figure 2(a)

Fig. 2 (a) Error signals in the time domain when the laser is stabilized to the cavity. Red: only with PZT, Blue: with the proposed method. (b) Frequency noise PSDs (in-loop). Black: shot noise floor, Grey: electronics noise floor. The peaks at the low frequency (100 Hz to 1 kHz) are caused by the power supply of the DFB EDFL.

Download Full Size | PDF

shows the error signals when the laser is stabilized to the cavity for the two cases, where only the PZT is used (blue), and where the PZT and the EOM are used together (red), respectively. By adding the fast FB, the residual error signal RMS fluctuations are reduced by more than a factor of 50. Figure 2(b) shows the corresponding frequency noise PSDs for these two cases. As mentioned earlier, the mechanical resonances of the PZT in the laser cavity limit the achievable feedback bandwidth to ~15 kHz. The blue trace shows the frequency noise PSD only with PZT feedback. The peak at the frequency around 17 kHz is the slow FB servo bump. By adding the proposed fast FB, the frequency noise PSD is significantly reduced as indicated by the red trace in Fig. 2(b). Noise components below 1 MHz are effectively suppressed as shown in the figure. The suppression is >10 dB at 100 kHz, ~30 dB at 10 kHz, and more than 40 dB at 1 kHz and below. The second servo bump appears at 1.8 MHz, which is an approximately 100-fold improvement over the PZT only approach. The 1.8 MHz servo bump is dominantly due to the phase lag of the LPF in front of the fast FB. The black and grey traces in Fig. 2(b) show the shot noise and the electronics’ noise floors. Note that the noise measurements in this paper are in-loop and thus the noise can be below the black and grey traces such as the red trace which is under 600 Hz (though the servo gain must be tailored such that this avoids adding noise to the out-of-loop signals). The slope of the noise attenuation factor with the EOM feedback loop (~10 dB/decade) at the lower Fourier frequency (below 2 kHz) is due to the double integration effect of our loop filter. Some readers, especially for researchers in the field of photonic microwave generation, may be familiar with phase noise PSD. The stabilized phase noise PSD is −71 dBc/Hz at 10 kHz offset in the optical domain (192 THz), which corresponds to −157 dBc/Hz for a 10-GHz microwave signal. The integrated RMS phase noise is suppressed from 124 to 2.2 mrad (integrated from 100 Hz to 1 MHz). It should be noted that this is an in-loop measurement, as the PDH method does not provide a simple out-of-loop measurement. At the level described here, the out-of-loop phase noise will be entirely limited by the stability of the reference cavity, in particular, the thermal noise (i.e. Brownian motion) of the mirror coatings. For a thorough noise analysis, the out-of-loop measurement is required. Note that, the residual amplitude modulation (RAM) caused by the control signal (DC-7 MHz) might be an issue. The RAM could convert to the frequency noise. Such noise would show up in an out-of-loop measurement but would remain hidden for the here-shown in-loop signals.

4. Conclusion

We demonstrated a relatively simple and efficient residual noise reduction method for a laser locked to an optical cavity via the PDH method. The method does not incur additional optical losses, nor does it require any changes to the optical configuration of existing PDH setups. It rather is an addition of a separate electronic feedback path, which greatly increases the feedback bandwidth and the loop gain. Here we showed a reduction of the residual RMS noise of the pre-stabilized laser by more than a factor of 50. This corresponds to a suppression of the weighted residual noise power by more than a factor of 3000 across the frequency range from 100 Hz to 2 MHz. We further showed that the addition of this extra feedback loop enables close to shot noise limited performance in conjunction with high finesse cavities (here ~400,000) and low optical power levels (~25 μW incident power). We foresee that this method can oftentimes eliminate the need of an external AOM – independent of the laser system used.

Funding

National Science Foundation (NSF CAREER Award, 1253044); DARPA (STTR W31P4Q-18-C-0002).

Acknowledgments

The Authors would like to acknowledge Dr. Mark Notcutt from Stable Laser Systems for loaning the High-F cavity and helpful discussions, and Lucas Kolanz for careful English proofreading. M. Endo is supported by Japan Society for the Promotion of Science, Fellowships for Research Abroad. We also gratefully acknowledge financial support from the National Science Foundation (NSF CAREER Award, 1253044); and from DARPA (STTR W31P4Q-18-C-0002).

References

1. R. X. Adhikari, “Gravitational radiation detection with laser interferometry,” Rev. Mod. Phys. 86(1), 121–151 (2014). [CrossRef]

2. C. Lisdat, G. Grosche, N. Quintin, C. Shi, S. M. Raupach, C. Grebing, D. Nicolodi, F. Stefani, A. Al-Masoudi, S. Dörscher, S. Häfner, J. L. Robyr, N. Chiodo, S. Bilicki, E. Bookjans, A. Koczwara, S. Koke, A. Kuhl, F. Wiotte, F. Meynadier, E. Camisard, M. Abgrall, M. Lours, T. Legero, H. Schnatz, U. Sterr, H. Denker, C. Chardonnet, Y. Le Coq, G. Santarelli, A. Amy-Klein, R. Le Targat, J. Lodewyck, O. Lopez, and P. E. Pottie, “A clock network for geodesy and fundamental science,” Nat. Commun. 7, 12443 (2016). [CrossRef] [PubMed]

3. S. Doeleman, T. Mai, A. E. E. Rogers, J. G. Hartnett, M. E. Tobar, and N. Nand, “Adapting a cryogenic sapphire oscillator for very long baseline interferometry,” Publ. Astron. Soc. Pac. 123(903), 582–595 (2011). [CrossRef]

4. F. N. Baynes, F. Quinlan, T. M. Fortier, Q. Zhou, A. Beling, J. C. Campbell, and S. A. Diddams, “Attosecond timing in optical-to-electrical conversion,” Optica 2(2), 141–146 (2015). [CrossRef]

5. T. M. Fortier, M. S. Kirchner, F. Quinlan, J. Taylor, J. C. Bergquist, T. Rosenband, N. Lemke, A. Ludlow, Y. Jiang, C. W. Oates, and S. A. Diddams, “Generation of ultrastable microwaves via optical frequency division,” Nat. Photonics 5(7), 425–429 (2011). [CrossRef]

6. X. Xie, R. Bouchand, D. Nicolodi, M. Giunta, W. Hänsel, M. Lezius, A. Joshi, S. Datta, C. Alexandre, M. Lours, P.-A. Tremblin, G. Santarelli, R. Holzwarth, and Y. Le Coq, “Photonic microwave signals with zeptosecond-level absolute timing noise,” Nat. Photonics 11(1), 44–47 (2017). [CrossRef]

7. S. L. Campbell, R. B. Hutson, G. E. Marti, A. Goban, N. Darkwah Oppong, R. L. McNally, L. Sonderhouse, J. M. Robinson, W. Zhang, B. J. Bloom, and J. Ye, “A Fermi-degenerate three-dimensional optical lattice clock,” Science 358(6359), 90–94 (2017). [CrossRef] [PubMed]

8. N. Hinkley, J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. Ludlow, “An atomic clock with 10(-18) instability,” Science 341(6151), 1215–1218 (2013). [CrossRef] [PubMed]

9. W. Zhang, J. M. Robinson, L. Sonderhouse, E. Oelker, C. Benko, J. L. Hall, T. Legero, D. G. Matei, F. Riehle, U. Sterr, and J. Ye, “Ultrastable silicon cavity in a continuously operating closed-cycle cryostat at 4 K,” Phys. Rev. Lett. 119(24), 243601 (2017). [CrossRef] [PubMed]

10. J. Davila-Rodriguez, F. N. Baynes, A. D. Ludlow, T. M. Fortier, H. Leopardi, S. A. Diddams, and F. Quinlan, “Compact, thermal-noise-limited reference cavity for ultra-low-noise microwave generation,” Opt. Lett. 42(7), 1277–1280 (2017). [CrossRef] [PubMed]

11. E. D. Black, “An introduction to Pound–Drever–Hall laser frequency stabilization,” Am. J. Phys. 69(1), 79–87 (2001). [CrossRef]

12. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983). [CrossRef]

13. T. C. Briles, D. C. Yost, A. Cingöz, J. Ye, and T. R. Schibli, “Simple piezoelectric-actuated mirror with 180 kHz servo bandwidth,” Opt. Express 18(10), 9739–9746 (2010). [CrossRef] [PubMed]

14. D. Goldovsky, V. Jouravsky, and A. Pe’er, “Simple and robust phase-locking of optical cavities with > 200 KHz servo-bandwidth using a piezo-actuated mirror mounted in soft materials,” Opt. Express 24(25), 28239–28246 (2016). [CrossRef] [PubMed]

15. T. Hausmaninger, I. Silander, and O. Axner, “Narrowing of the linewidth of an optical parametric oscillator by an acousto-optic modulator for the realization of mid-IR noise-immune cavity-enhanced optical heterodyne molecular spectrometry down to 10⁻¹⁰ cm⁻¹ Hz^−1/2,” Opt. Express 23(26), 33641–33655 (2015). [CrossRef] [PubMed]

16. D. Shoemaker, A. Brillet, C. N. Man, O. Crégut, and G. Kerr, “Frequency-stabilized laser-diode-pumped Nd:YAG laser,” Opt. Lett. 14(12), 609–611 (1989). [CrossRef] [PubMed]

17. E. Benkler, F. Rohde, and H. R. Telle, “Endless frequency shifting of optical frequency comb lines,” Opt. Express 21(5), 5793–5802 (2013). [CrossRef] [PubMed]

18. R. Kohlhaas, T. Vanderbruggen, S. Bernon, A. Bertoldi, A. Landragin, and P. Bouyer, “Robust laser frequency stabilization by serrodyne modulation,” Opt. Lett. 37(6), 1005–1007 (2012). [CrossRef] [PubMed]

19. D. Gatti, R. Gotti, T. Sala, N. Coluccelli, M. Belmonte, M. Prevedelli, P. Laporta, and M. Marangoni, “Wide-bandwidth Pound-Drever-Hall locking through a single-sideband modulator,” Opt. Lett. 40(22), 5176–5179 (2015). [CrossRef] [PubMed]

20. M. Musha, K. Nakagawa, and K. Ueda, “Wideband and high frequency stabilization of an injection-locked Nd:YAG laser to a high-finesse Fabry Perot cavity,” Opt. Lett. 22(15), 1177–1179 (1997). [CrossRef] [PubMed]

21. X. Xue, F. C. O. Leo, Y. Xuan, J. A. Jaramillo-Villegas, P.-H. Wang, D. E. Leaird, M. Erkintalo, M. Qi, and A. M. Weiner, “Second-harmonic-assisted four-wave mixing in chip-based microresonator frequency comb generation,” Light Sci. Appl. 6(4), e16253 (2017). [CrossRef]

22. E. Benkler, F. Rohde, and H. R. Telle, “Endless frequency shifting of optical frequency comb lines,” Opt. Express 21(5), 5793–5802 (2013). [CrossRef] [PubMed]

23. P. C. D. Hobbs, Building Electro-Optical Systems: Making It All Work, 2nd ed , Vol. 71 of Wiley Series in Pure and Adaptive Optics (John Wiley & Sons, 2011).

Residual phase noise suppression for Pound-Drever-Hall cavity stabilization with an electro-optic modulator

Abstract

1. Introduction

2. Method

3. Experiment and results

4. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (2)

Equations (7)

OSA Continuum