High-frequency broadband laser phase noise cancellation using a delay line

: Laser phase noise remains a limiting factor in many experimental settings, including metrology, time-keeping, as well as quantum optics. Hitherto this issue was addressed at low frequencies ranging from well below 1 Hz to maximally 100 kHz. However, a wide range of experiments, such as, e.g., those involving nanomechanical membrane resonators, are highly sensitive to noise at higher frequencies in the range of 100 kHz to 10 MHz, such as nanomechanical membrane resonators. Here we employ a fiber-loop delay line interferometer optimized to cancel laser phase noise at frequencies around 1.5 MHz. We achieve noise reduction in 300 kHz-wide bands with a peak reduction of more than 10 dB at desired frequencies, reaching phase noise of less than − 160 dB(rad 2 /Hz) with a Ti:Al 2 O 3 laser. These results provide a convenient noise reduction technique to achieve deep ground-state cooling of mechanical motion.


Introduction
Lasers are nowadays well-established as the workhorse of modern telecommunication, metrology, as well as developing quantum technologies.However, as has been already realized by Schawlow and Townes [1], fundamentally the phase noise of a laser is finite and fundamentally limited.Practically, many more effects, such as thermal [2] or acoustical [3] noise of laser cavities, contribute to the total phase noise.Many techniques are employed to reduce phase noise, and thus the linewidth of lasers.To great success, external ultra-stable cavities have been employed as references [4,5], achieving linewidths well below 1 Hz, but broadband phase noise remains a problem in many applications.In contrast to intensity noise, which usually exhibits localized peaks due to relaxation-oscillation [6], phase noise in lasers tends to exhibit both a broadband noise floor and technical noise peaks.
As an alternative method to reference cavities, an unbalanced Mach-Zehnder interferometer with an optical fiber delay line in one of its arms can be used.Such setups are routinely employed to characterize laser linewidths [7].Remarkably, one can also use the signal of such a delay-line setup to feedback on the laser light's phase, thereby reducing its noise [8][9][10][11][12][13][14][15][16][17].All previous approaches operated in a relatively low-frequency range, starting from very low infrasonic frequencies up to maximally around 100 kHz.
At low frequencies, acoustic isolation of fibers has been successfully implemented [17], but at higher frequencies the thermomechanical and thermoconductive noises are unavoidable [18].Fiber noise also poses limitations on transfer of optical frequency standards, and thus a reverse approach is often employed in which the fiber is stabilized to a narrowband stable laser [19,20].Several works have also studied fundamental limits to fiber noise and related fiber strain sensing [21][22][23].
Here we employ a 50-meter-long fiber delay line to reduce laser phase noise at high frequencies.In particular we are interested in reducing phase noise at frequencies in the vicinity of 1.5 MHz that correspond to the resonance frequency of a membrane mechanical oscillator.Using a feedback loop, with the gain concentrated around the frequency of interest, we achieve phase noise of −164 dB(rad 2 /Hz) at 1.5 MHz frequency offset of a Titanium-Sapphire laser (M-Squared SolsTiS), providing noise reduction in a previously untackled frequency regime with a very high absolute bandwidth.The results are uniquely enabled by shot-noise limited detection at high light powers, combined with an optimized fiber length.
Noise at such high frequency offsets is an essential limitation in quantum optomechanics, where the sideband at the resonance frequency of the oscillator leads to significant classical drive forces [24][25][26].We envisage that using this light for feedback cooling of membrane resonators [27][28][29] in a sideband-resolved cavity quantum optomechanics regime will allow to break an important barrier of 0.1 residual occupation of phonons of a membrane resonator mode at liquid helium temperatures (∼ 4 K).
The paper is organized as follows.We first introduce the experimental setup, which involves two similar unbalanced interferometers, for feedback and characterization.Next, we introduce the model for detection of the laser's phase noise, including a treatment of spurious fiber noise and photon shot noise, as well as a model for feedback.Finally, we show results for noise suppression in several MHz-level bands and compare them with our model prediction.

Setup
As presented in Fig. 1, our test experiment consists of two unbalanced Mach-Zehnder interferometers with the short arm being 1 m-long and the long arm being 50 m-long.All fiber in our setup is polarization maintaining (PM), single-mode fiber (PM-780HP), which allows us to maintain good interometric visibility and polarization stability.We inject equal amounts of light into both arms using a 50:50 fiber beamsplitter.In each interferometer we place an additional piezo-actuated mirror for stabilization of low-frequency phase drifts in order to lock the interferometer at the optimal (balanced) point for sensing phase noise.The beams are combined on a polarizing beamsplitter (PBS), and then sent through an additional half-wave plate and another PBS onto a balanced detector.
Our detectors feature a high quantum efficiency of  ≈ 0.9 and separate AC/DC differential outputs.The low-frequency DC output is used to actively stabilize the interferometer with the piezo-actuated mirror.In setup 1, the high-frequency AC output is sent to a STEMlab Red Pitaya 125-14 board equipped with PyRPL software [30], which implements the feedback filter in a field-programmable gate array (FPGA), and drives a free-space EOM (electro-optic modulator) through an additional 20 dB attenuator to minimize electronic and quantization noises.The signal is also sent to an FFT spectrum analyzer in order to measure in-loop noise.
The feedback is achieved using an I/Q modulator/demodulator module in the RedPitaya/PyRPL architecture.We first mix the error signal with a carrier wave at a desired frequency  c , apply a lowpass filter which gives the bandwidth  of the filter, and finally modulate the signal back with the carrier wave at  c .The phase of the carrier wave determines phase  of our feedback, which is empirically optimized.The signal is also multiplied by a certain gain factor .
In setup 2, we send the AC signal to a spectrum analyzer for out-of-loop noise measurement.It uses an additional fiber EOM which can be used for absolute phase-noise calibration as it features a flat, calibrated response to phase modulation.
Our laser operates at  = 852 nm and outputs approximately 200 mW of light power.We send 120 mW through the free-space EOM for the entire experiment [29], part of which is sent to both fiber delay line setups.In setup 1, we obtain P = 11 mW of total power impinging on the balanced detector, which allows for fiber-noise limited measurement of laser phase noise, increasing the signal-to-shot-noise ratio.For setup 2, we use P = 5 mW of total power at the Fig. 1.Schematic of the delay line feedback loop (setup 1) and an additional loop for out-of-loop measurement (setup 2).At the input, the interferometers feature a fiber 50:50 beamsplitter, while at the output the beams are combined via free-space polarization optics and sent onto a balanced photodetector.Both interferometers are independently stabilized with piezo mirrors at low frequencies, such that the detection remains balanced.Feedback to the laser light is applied from setup 1 via an electronic controller through a free-space EOM.An additional fiber EOM is used for absolute calibration of setup 2. detector.

Detection
We consider input laser light with a fluctuating phase given by ().The phase of the field at the output of the long delay is given by ( − ) +   (), where   ( − ) is the delayed phase signal due to propagation through the long arm, while at the output of the short arm we simply have ().At the output of the interferometer we will measure the phase difference given by () − ( − ).The frequency-domain transfer function for the laser phase noise is therefore given by: with delay  = /, introduced by the fiber of length  with refractive index  and the speed of light in vacuum .Simultaneously, the frequency-domain transfer function for the fiber noise is unity.
The fiber noise is added independently as a detector noise term together with shot noise.Active stabilization of the interferometer arms' relative phase preserves power balance, such that we can assume a total power P impinging on the balanced differential photodetector to be equally split between the individual diodes.In this configuration, the total power spectral density (PSD) of the registered optical signal is: where  is the quantum efficiency of the detector.The first term represents the laser phase noise and the second term represents fiber noise.Finally, photon shot noise PSD is given by 2ℎ P with  = / and ℎ being the Planck's constant.
Taking the transduction to/from phase noise into account, we may express the noise in terms of equivalent laser phase noise: From this, we quite clearly observe that using a high optical power P diminishes the shot-noise contribution.
For the sake of this model, let us assume that the two delay line setups (in-loop and out-of-loop) have identical powers and efficiencies.Conveniently, we can observe the laser phase noise directly in the cross spectral density (CSD) between the two setups: This is possible as the laser phase noise is the only correlated noise shared between the two setups.In practice, we use this method to identify the laser phase noise contribution to total measured noise.

Fiber noise
In order to estimate the noise of the fiber, we use the theory of Duan [18] and notice that at our frequencies of interest, the noise is heavily dominated by the thermoconductive noise, i.e. fluctuations of temperature transduced to phase fluctuations via thermal expansion and temperature dependence of the refractive index.The expected fiber noise is: where   is the Boltzmann constant,  is temperature,  is thermal conductivity, d d is thermooptic coefficient,  is coefficient of linear expansion,  is thermal diffusivity,  0 is mode power profile radius and  1 represents the exponential integral.

Feedback
We now introduce feedback, such that in the Fourier domain: where ζ represents shot noise with PSD of 2ℎ P, while φ and φf represent stochastic laser and fiber noises.The feedback transfer function  (  ) shall include both our desired feedback, as well as an undesired but unavoidable delay given by a prefactor  −2   D  , mostly coming from the electronic processing delay ( D ≈ 250 ns).The feedback acts on light "shared" by both interferometers.In this case, setup 1 is the in-loop setup used for feedback, while setup 2 serves for independent out-of-loop characterization.
As a result of the feedback, the actual laser phase noise becomes: We see that the original phase noise is suppressed, but new noise is added due to detection and fiber noise in setup 1.This directly allows us to identify the out-of-loop detector (setup 2) noise  fb,meas   by substituting  fb   into Eq. 3 with uncorrelated fiber and shot noise.For the in-loop detector, however, the same detection noise becomes present in the feedback, which leads to interference.We obtain the following measured noise: where the first term is the expected reduced laser noise, while the second term represents the detection noise itself as well as its self-interference.In particular, the measured noise in this case may fall well below the original detection noise.
In practice, we employ the following feedback which results from the I/Q modulation/demodulation technique as described in Sec.2: where we control the following parameters: gain , phase , central frequency  c and bandwidth .In the experiment we use an additional fiber EOM with   = 4.65 V to recover an experimental profile of phase sensitivity curve of setup 2. Simultaneously, we may apply a calibration signal to the free-space EOM, which will be detected by both setups, and obtain a relative calibration Total incoherent noise without shot noise (setup 1) Theoretical fiber noise Fig. 3. Incoherent part of noise in setup 1 with shot noise subtracted (blue part in 2) compared with theoretical prediction for noise of the delay fiber.The result is normalized by the equivalent fiber noise per unit fiber length, where we assume that all measured incoherent noise is fiber noise of setup 1 with respect to calibrated setup 2. For the calibration we apply a low-frequency square-wave pattern and compare amplitudes of its Fourier components as registered by both setups.

Experimental results
Using such calibration, we can measure the cross-spectral density (CSD) which only contains the common-mode laser phase noise (Eq.4).After expressing all PSDs and the CSD in equivalent laser phase noise units (using calibrated phase-modulation responses), we identify the incoherent (uncorrelated) parts of noise in both setups.Finally, we also measure the photon shot noise by subsequently blocking arms of the interferometer and adding resulting registered noise.
Figure 2 presents the raw measured detector noise (expressed in shot-noise units) and equivalent laser phase noise for setup 1.We decompose the noise into the coherent (correlated) laser phase noise and the incoherent detection noise part, which is itself composed of shot noise, fiber noise, and other detection noise.While we expect that the incoherent noise is heavily dominated by fiber noise, our analysis does not require this assumption.With  = 250 ns ( = 1.5,  = 50 m) we observe maximum sensitivity to phase noise at a 2 MHz offset, and minima of sensitivity at 0 MHz and 4 MHz.In terms of equivalent laser phase noise, this means that in the range above 3.5 MHz, where laser phase noise is becoming smaller, we are heavily dominated by detection noise.Below this frequency, however, our detection setup exhibits a signal-to-noise ratio of at least 1.By using an optical power of P = 11 mW, we make sure that at our main frequency of interest at 1.5 MHz, detection noise is dominated by fiber noise and not shot noise.Large laser phase noise peaks around 2.3 MHz make the subtraction procedure sensitive to uncertainties, and thus in order to avoid imprecision we interpolate the detection noise by a second order polynomial in the 2 MHz-2.5 MHz region.
With the extracted incoherent noise of setup 1, we may also compare it with a prediction for fiber noise from Eq. 5.The following parameters are assumed [18,31]: s and  0 = 1.5 µm.In Fig. 3 observe that while the general behavior is well reproduced, the predicted fiber noise lies slightly below our measured incoherent detection noise (with shot noise subtracted).We attribute this discrepancy either to additional noise, or more likely to inaccuracy in used parameters, which are not directly available for the PM-780HP fiber we use, but are rather extracted from general material properties.
Next, we proceed to apply feedback and observe the noise registered in both setups.With other parameters optimized, we use a filter with  c = 1.5 MHz and bandwidth  = 78 kHz while changing the loop gain from  = 1.2 through  = 6 to  = 12.In the top panel of figure 4 we show the laser phase noise as measured with the out-of-loop setup 2.Here we subtract the uncorrelated incoherent noise.We observe that with our feedback we register significantly less noise with up to 10 dB of reduction for the highest gain (see the bottom row).A phase noise peak at 1.54 MHz is suppressed to an even greater degree.Simultaneously, due to phase mismatch we observe increased noise away from the central frequency.For the purpose of noise reduction at the specific frequencies of our optomechanical experiment, this does not pose a problem.
In the middle row we show the simultaneously registered noise of the in-loop setup 1.This noise is the raw noise of the detector, converted into equivalent laser phase noise units.Here we observe that the noise, particularly for the highest gain, reaches well below the detection noise.This noise squashing behavior is expected as the result of noise self-interference.Our experimental results are accompanied by theory curves that use the model from Sec. 3 along with measured responses and detection noise.We observe particularly good agreement for the most important phase noise measured by the out-of-loop detector.A slightly higher noise than expected is registered in the in-loop detector, which may be a result of additional electronic detection noise.
Finally, we apply the feedback at different central frequencies, as shown in Figs. 5 and 6.We observe up to 12 dB of noise reduction and bandwidths of at least 300 kHz in all cases.

Conclusions and prospects
We have demonstrated the operation of the fiber delay line active laser phase noise reduction setup in the MHz frequency range, which lies outside of previously explored frequency domains.We demonstrated phase noise reduction of at least 10 dB and broadband operation, achieving very low absolute phase noise densities at the level of approximately −160 dB(rad 2 /Hz).We have also demonstrated an effective model for predicting the performance of our method, as well as a convenient way to identify noise components in the system based on cross-spectral density evaluation.Our results also include measurements of the fiber noise at high frequencies, which show reasonable agreement with predicted thermoconductive noise.
Our results can find particular applications in optomechanics, where reduction of noise around a specific offset frequency is desired.This applies to both sideband and feedback (cold-damping) cooling of trapped particle oscillators [24,32,33], room-temperature integrated resonators [34], membrane-in-the-middle systems [28,29] and others [25,26].Other applications sensitive to noise at a particular offset frequency include the driving of Raman transitions [?, ?] or hybrid electro-opto-mechanical converters [35].
Our main prospective application is generation of low-phase-noise light for optomechanical sideband cooling.As estimated by Kippenberg et al. [25], the limit for the expected final phonon occupation (i.e. with intracavity power and detuning optimized) for an optomechanical resonator in the sideband-resolved regime due to phase noise is given approximately by: It is also worth mentioning an alternative approach to reduce noise at MHz frequencies.This approach makes use of a filtering cavity, which has been also demonstrated as a method to measure laser phase noise [25].In such setup, one uses light directly transmitted through a narrowband cavity.However, special treatment would be required to make sure that mirror thermal noise, which is widely considered an important limitation of optomechanical setups [36], remains well below the input laser phase noise.We estimate that such cavity would either have to be sufficiently long to reduce mirror noise transduction, or cryogenically cooled to suppress thermal motion.
Finally, we envisage that the delay line approach can be further improved by reducing fiber thermal noise.One approach would be to embed the delay line in a cryogenic environment.We estimate that due to simultaneous reduction of thermal noise and the thermorefractive coefficient, the fiber noise could be reduced by more than 20 dB at moderate liquid-nitrogen temperatures.Furthermore, the noise can be greatly reduced with an increased mode field diameter.For a given wavelength, this could be possibly achieved in large-mode-area fibers.

Fig. 2 .
Fig. 2. Components of noise registered by the balanced detector in setup 1.(a) Optical power spectral density normalized to photon shot noise and (b) the same signal in terms of equivalent laser phase noise as calculated from transduction of phase noise into detected noise.The three components are photon shot noise, detection noise (primarily composed of fiber noise) and the signal which corresponds to laser phase noise.The peak sensitivity due to delay loop transduction is obtained in the vicinity of 2 MHz.

Fig. 4 .
Fig.4.Laser phase noise for loop parameters  c = 1.5 MHz and bandwidth  = 78 kHz and  = 1.2, 6 and 12 for columns from left to right, respectively.The first row presents the laser phase noise as measured by out-of-loop setup 2, with detection noise subtracted.In the second row we show the in-loop noise of setup 1, converted to equivalent laser phase noise units.Bottom row shows the reduction of laser phase noise, as inferred from the out-of-loop measurement.

Fig. 5 .Fig. 6 .
Fig. 5. Laser phase noise as measured by out-of-loop setup 2 and respective noise reduction due to feedback with loop parameters:  c = 0.8 MHz,  = 78 kHz and  = 10.Red curve is the original noise, blue represents the noise after feedback, and yellow curve is the theoretical prediction.