1 Introduction

Because of their high spectral purity and frequency stability, ultra-stable oscillators at optical frequencies have been used in a variety of research fields such as optical frequency standards [1–6], high-precision spectroscopy [7–14], interferometric gravitational-wave (GW) detection [15–21], and the test of Lorentz Invariance (LI) using optical resonators [22–27]. Unprecedented frequency stability has been achieved with the continuous pursuit of ultra-stable lasers employing high-finesse optical cavities [28–37]. A relatively compact system can reach 1 × 10−15 frequency stability on timescales of 1–100 s [38–45], and this level of stability is extended to longer time up to 1,000 s [46–48]. The frequency instability of modern cavity-stabilized lasers has been steadily pushed down to 10−16 [49–51] and even to the realm of 10−17 [52] due to the reduced thermal noise [53] of the optical cavity.

As several measures have been proved to be effective in reducing the thermal noise [49, 52, 54] and additional methods are in consideration [55, 56], residual amplitude modulation (RAM), a side effect of phase modulation using an electro-optic (EO) crystal, has the potential to be one of the major sources affecting the frequency stability of ultra-stable optical oscillators. Early and follow-up investigations have already confirmed its prominent origin being the birefringence of EO crystal [57], along with other mechanisms [58–64]. Quantitative analyses have been performed to address the impact of RAM on various applications [46, 60–62, 65–68]. Being able to extract the information of RAM actually experienced by the Pound–Drever–Hall (PDH) discriminator [28] is critical to a successful control of RAM in ultra-stable optical oscillators. High fidelity can be achieved by extracting RAM inside the PDH-frequency-locking path [69]. Another method that has been explored in great detail is sensing RAM in a separate path with relatively simple configuration [57, 62, 70]. Nevertheless, the effectiveness of RAM control using a separate path in PDH frequency locking remains to be confirmed because of the caution that any parasitic etalon and inhomogeneity, or the optical cavity itself may degrade or destroy the coherence between the sensing and locking paths, leading to a distorted representation of RAM [71].

We have developed two 1,064-nm Nd:YAG lasers stabilized by ultra-stable optical cavities to experimentally verify the effectiveness of the RAM control and investigate various noise mechanisms in ultra-stable optical oscillators. Active control of RAM is implemented by a separate sensing path, and an extra EO crystal is inserted in front of the electro-optic modulator (EOM) to increase the control bandwidth. With the laser tuned off resonance, the performance of this control scheme is first evaluated by comparing residual noise spectra in the sensing and control paths. Furthermore, the birefringence-induced RAM in the presence of an optical cavity is theoretically analyzed and an in situ verification is performed by applying harmonic perturbations of RAM on the locked laser and measuring the rejection ratios both in sensing path and in optical heterodyne beat between the perturbed laser and a reference laser. In addition, laser frequency noises originated from vibration, quantum-limited shot noise, and electronic noise are measured and analyzed, providing detailed information of noise properties that is beneficial to the development of next-generation ultra-stable optical oscillators as well as related applications in which frequency modulation plays an important role. We also experimentally verify a simplified relationship [72] between the linewidth and the underlying noise that modulates the laser frequency.

We proceed to first give a brief description of the frequency-stabilized Nd:YAG lasers in Sect. 2. Section 3 introduces the experimental setup of the active RAM control and the verification of its effectiveness. Section 4 details the measurement and analysis of the frequency noises of different origins. The frequency stability and the spectral linewidth of the optical oscillators are also examined in this section. Conclusions are provided in Sect. 5.

2 Experimental setup of two ultra-stable Nd:YAG lasers

We have developed two similar but independently operated ultra-stable optical oscillators, each of which consists of a 1,064-nm diode-laser-pumped Nd:YAG laser stabilized by a high-finesse optical cavity using PDH-frequency-locking method [28]. Figure 1 shows the experimental setup for one of the two systems, which are designated as the WEST and EAST systems. The optical cavity consists of a 100-mm spacer and two ultra-low-loss mirrors all made of ultra-low expansion glass (ULE). The two mirrors, one plano and one concave (0.5 m in radius of curvature), are optically contacted to the end surfaces of the spacer, which has an installation flange on the middle and is tapered toward two ends [73]. By using cavity ring-down technique, the finesses of the cavities are determined to be 150,000 (WEST) and 148,000 (EAST) at 1,064 nm. Inside a cylindrical vacuum chamber, the optical cavity is vertically supported by three Teflon posts at a vertical position that is 2.3 mm below the horizontal midplane of the spacer to reduce the vibration sensitivity of the optical cavity [38–42, 55, 73, 74]. The vacuum chamber and the surrounding optics are mounted on a passive vibration isolation platform. The laser beam is transmitted to the vibration isolation platform through a 3-m polarization-maintaining (PM) fiber. To suppress the phase noise introduced by the optical fiber, we adopt a cancellation scheme [75] in which the reflection from the fiber end on the cavity side is used to generate the correction signal.

Fig. 1
figure 1

Experimental setup of the two cavity-stabilized Nd:YAG lasers. a Fiber noise cancellation and heterodyne-beat measurement. b PDH-frequency-locking module placed inside a hermetically sealed and temperature controlled enclosure. λ/2 half-wave plate, λ/4 quarter wave plate, GT Glan–Taylor prism, Col. collimator, EOM electro-optic modulator, ISO optical isolator, PD photodetector, PBS polarization beam splitter, DBM doubly balanced mixer, FR (45°) 45° Faraday rotator, AOM acousto-optic modulator, FNC fiber noise cancellation

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Two stages of temperature control are employed to reduce the thermal fluctuation experienced by the optical cavity. The entire subsystem, including the vibration isolation platform and all PDH-locking components sitting on the platform, is placed inside a hermetically sealed aluminum enclosure whose six plates are independently temperature controlled with stability of 10 mK. Long-term temperature stability better than 1 mK is achieved by the second-stage analog controllers that independently stabilize the temperatures of the vacuum chamber and the ion pump. The acoustic isolation relies primarily on the aluminum enclosure, which, when hermetically sealed, provides (30–100)-fold attenuation of sound pressure measured from 10 to 100 Hz.

The EAST (WEST) laser is phase modulated at 12.288 MHz (10 MHz) with modulation index of ~1. We guide 140 μW of laser power to each optical cavity that has a reflection contrast about 10 %. The PDH error signal is amplified by a loop filter and then sent to a piezoelectric transducer clamped on the laser resonator for correcting fast frequency fluctuation. Frequency correction with large dynamic range is provided by an auxiliary control loop that operates by adjusting the temperature of the laser resonator. The optical intensity is monitored by using the DC output of PD1, and it is stabilized by changing the radio-frequency (RF) driving power of the AOM with a voltage-controlled attenuator (VCA). By altering the incident laser power the intensity-frequency dependence of the two cavities is determined to be 2 Hz/μW in respect to the incident laser power of 140 μW. When the laser intensity is stabilized, the contribution of the laser intensity noise is nearly two orders of magnitude lower than the current frequency instability. The AC output of PD1 is used for the active control of RAM, which will be detailed in the following section (Sect. 3).

3 Active control of RAM

3.1 Control scheme

We adopt and experimentally verify an active control scheme [57, 62, 70] that detects RAM in a separate path and feeds back the correction signal to the phase modulator to suppress RAM. The experimental setup can be found in Fig. 1b and Fig. 2 depicts in detail the control loops involved in the process. Before entering the optical cavity, the phase-modulated laser is split into two beams for PDH frequency locking and detection of RAM. Large dynamic range is covered by a slow control loop that adjusts the temperature of the EOM crystal [57, 62, 76]. To improve high-frequency response, a fast control loop is introduced by inserting a small-size EO crystal [magnesium-doped lithium niobate (MgO·LiNbO3)] in front of the original EOM crystal. This newly added one is shorter (5 mm) compared with the 35-mm EOM crystal but has the same cross section of 3 mm × 5 mm. The correction signal is amplified by a high-voltage (HV) amplifier and then applied to the electrodes of the small crystal with the resulting electric field being along the z axis (c.f. Figure 2). Alternatively, in the case of small half-wave voltage such as a waveguide-type modulator, a bias tee can be used to combine the DC voltage with the RF signal and a single crystal can be used [57, 70].

Fig. 2
figure 2

Feedback loops used for active control of RAM. A small-size EO crystal (cross section: 5 mm × 3 mm, length: 5 mm) in front of the EOM performs fast compensation. A slow loop provides large dynamic range by exploring the temperature-dependent crystal birefringence. Harmonic perturbations are added at the input of the HV amplifier to measure the rejection ratio of RAM (see text for details). TEC thermoelectric cooler, P proportion, I integration, Temp. ctrl. temperature controller, HV high-voltage amplifier

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To achieve enough discrimination gain for the feedback control, the polarization of the laser beam is deviated from the z axis by a small oblique angle of ~3°. Prior to the active control, RAM is minimized according to procedures detailed in a previous work [62] and its fluctuation is greatly reduced by housing the setup with hermetically sealed enclosure. After these adjustment and optimization, the level of RAM measured at the AC output of PD1 is −50 dBm, but it can still drift to value as large as −45 dBm on a timescale of 10 h. The level is reduced to −52 dBm and remains stable when the active control of RAM is engaged.

Crucial to the successful detection of RAM is to avoid various RF interferences such as those from AOM and EOM drivers. Band-pass filters are inserted after PD1 and PD2, and low-noise shielded twisted pairs are used to send the demodulated signals from the doubly balanced mixers to the succeeding operational amplifiers. Moreover, to minimize the influence of the differential grounding potentials, the IF ports of the doubly balanced mixers do not share the local grounding pads with the RF and LO ports, and the inputs of the first-stage high-gain operational amplifiers are floated.

3.2 Noise spectra and instabilities of RAM in two separate paths

The approach of a separate detection path is first examined by measuring the noise spectra in the two paths with laser steered away from the cavity modes, meaning that the cavity only acts as a close-to-perfect reflector. For a phase-modulated laser beam with imprints of various RAM-related mechanisms, the subsequent photodetection yields a photo current \(I = I_{0} \left[ {1 + \varepsilon \left( t \right)\cos \left( {\omega_{m} t + \phi } \right)} \right]\) with harmonics being omitted, where ω m is the modulation frequency, I 0 is a DC current, ɛ(t) is the amplitude modulation index, and ϕ is a phase shift. Time series of the demodulated signals in two paths each span about 5 h are recorded by a data acquisition card (NI 4462) and then converted to noise spectra and Allan deviations.

Figure 3 shows the noise spectra and the Allan deviations of the RAM-induced frequency drift, along with those of the modulation index ɛ(t). Noise components resulting from RAM are observed below 0.1 Hz with only temperature stabilization of EOM crystal. When active RAM control is implemented in the sensing path, out-of-loop noise in the path intended for frequency locking is also suppressed for Fourier components <0.1 Hz. In-loop noise in the sensing path reveals a control bandwidth up to 700 Hz, and it approaches the noise floor introduced by the data acquisition card.

Fig. 3
figure 3

Verification of the effectiveness of active control of RAM with laser tuned off resonance with the optical cavity. a Noise spectra and b Allan deviations of RAM [modulation index ɛ(t)] and the corresponding frequency instabilities. The frequency noise and instability are estimated using a PDH discrimination slope of 1.25 × 10−5 V/Hz

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The effectiveness of the active RAM control is also apparent in the time domain representation with Allan deviation. Starting from 1 s, the level of RAM without active control monotonically increases from 3 × 10−7 by more than an order of magnitude to reach 1 × 10−5 at 1,000 s, whereas in the case of active control, it is suppressed to below 1 × 10−6 with a minimum of 2 × 10−7 at ~2 s. The corresponding RAM-induced frequency instability is below 1 × 10−15, estimated with a discrimination slope of 1.25 × 10−5 V/Hz in the current system.

Above 0.1 Hz, contributions from the shot noise and electronic noise of photodetectors surpass the RAM-induced frequency noise. With active RAM control, these noises produce a characteristic plateau in the noise spectrum of the PDH-locking path and an Allan deviation that is inversely proportional to τ 1/2 at averaging time 1 s. Small difference is observed between the RAM sensing and frequency-locking paths both in frequency and time domains, implying the possibility of some residual differential process between the two paths even in the absence of the leakage field from the optical cavity.

3.3 In situ verification

Additional analysis and measurements are performed to investigate whether an on-resonance optical cavity dramatically modifies the property of RAM. In PDH frequency locking, the imperfect alignment of the laser polarization with respect to the principal axis of the EO crystal results in a DC shift of the PDH error signal. If we assume a fast modulation (i.e., the modulation frequency is much higher than the resonant linewidth of the optical cavity) and only consider optical carrier and its first-order sidebands, the DC shift at the resonant frequency ω 0 can be approximated as

$$I_{\text{RAM}} = - 4ab\eta_{\text{photo}} E_{0}^{2} J_{1} (M)F_{r} (\omega_{0} ) \cdot \sin \varDelta \phi ,$$
(1)

where F r (ω) is the amplitude reflection coefficient of the optical cavity [77], a and b are misalignment factors [57], η photo is the responsibility of the photodetector, E 0 is the amplitude of the optical field, J 1(x) is the first-order Bessel function, M = δ e  − δ o is the difference in the modulation indices of the extraordinary and ordinary lights, and Δϕ is the phase shift arising from the natural birefringence of the EO crystal. Equation (1) indicates that the net effect of introducing an optical cavity is to add a scaling factor F r (ω 0) to the birefringence-induced RAM without the optical cavity [57]. Consequently, an on-resonance optical cavity does not undermine the coherence between the sensing and PDH-frequency-locking paths and hence the same amount of RAM suppression can be achieved in the PDH-frequency-locking path. It is also obvious that optical cavities with higher reflection contrast will have lower susceptibility to the birefringence-induced RAM.

This analysis is examined by introducing sinusoidal perturbations at the input of the HV amplifier to modulate the level of the RAM presented in the laser beam. The perturbation is amplified by the HV amplifier and then added to the EO crystal (c.f. Fig. 2), producing a variation of the beat note frequency with a root-mean-square amplitude of 160 Hz. Firstly, the laser is detuned from the resonance of the optical cavity and the measurement is performed out of loop in the path intended for the PDH frequency locking. Secondly, we lock the perturbed laser to the optical cavity and heterodyne beat this laser against the second laser that is locked to its own high-finesse optical cavity to serve as a stable reference. A frequency-to-voltage converter and a fast-Fourier-transformation (FFT) analyzer are used to extract RAM in the beat note. In both measurements, the active RAM control is switched on and off to allow for the determination of the rejection ratio.

Figure 4 plots the rejection ratios obtained in the two measurements. The two sets of rejection ratio, one inferred from the out-of-loop measurement with laser unlocked and the other observed in the optical frequency of the locked laser, agree with each other and show the −12 dB/octave slope that is in accord with the open-loop gain of the active RAM control. From this comparison and Eq. (1), we conclude that when RAM is caused primarily by the birefringence of the EO crystal the optical cavity has a negligible effect on the control of RAM using a separate sensing path.

Fig. 4
figure 4

Rejection ratios obtained with active RAM control. Harmonic perturbations of RAM are added at the input of the HV amplifier driving the EO crystal. Squares are measured out of the RAM-control loop with the test laser tuned off resonance (inset a). Circles are obtained using the optical heterodyne beat between the perturbed but locked laser and a stable reference laser (inset b). The two sets of rejection ratios agree with each other and also with the −12 dB/octave slope of the open-loop gain profile. In the second measurement, the laser is locked with relatively weak leakage field from the optical cavity, which has a reflection contrast of 10 %

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4 Analysis of frequency noise

4.1 Noise spectrum and frequency stability

The heterodyne beat of two locked Nd:YAG lasers at 927 MHz is measured by a counter (CNT-90) and subsequently converted to noise spectra and Allan deviations. During measurements, the intensity stabilization is activated and the RAM control is switched on and off for comparisons. Figure 5 plots the frequency noise of the beat note and the calculated thermal noise [53], together with individual contributions from RAM, shot noise, vibration, and electronic noise.

Fig. 5
figure 5

Frequency noise of two Nd:YAG lasers stabilized by high-finesse optical cavities and contributions of noises with different origins in the EAST system. Vibration-induced frequency noise is converted from ground vibration, using the experimentally determined vibration sensitivity of the optical cavity. The photodetection (PD) noise is measured with the EO modulation switched off

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The vibration-induced frequency noise is obtained from the vibration spectrum measured on the vibration isolation platform and the acceleration sensitivities of the optical cavities, which are 243 kHz/g (EAST) and 45.3 kHz/g (WEST), measured by vertically driving the floated vibration isolation platform with an electromagnet at 10 Hz. Contributions from the WEST and EAST systems are combined incoherently to give the noise spectrum shown in Fig. 5. The excess noise around 10 Hz in the vibration noise spectrum comes from the mechanical resonance of the optical table on which the vibration isolation platforms are installed.

The frequency noise in the beat note is close to the calculated thermal noise (0.24 · f −1/2) at Fourier frequencies from 0.01 to 2 Hz. Beyond 10 Hz, there is excess noise that is dominated by the quantum-limited shot noise and electronic noise from photodetectors. The two peaks around 3 Hz in the frequency noise are not caused by the vibration but their source is not clear at present. The shot noise [29] is calculated from the optical power of the cavity reflection and relevant experimental parameters. The electronic noise is measured at the outputs of photodetectors by blocking the laser beams. Additional contributions due to active RAM control have been included in the analysis of shot noise and the electronic noise of the photodetectors.

Figure 6 is the frequency stabilities of the optical heterodyne beat between two cavity-stabilized Nd:YAG lasers. The Allan deviation is computed after a linear drift is removed from the beat note. Note that a varying drift rate is observed in the beat note due to the thermal drift of the optical cavities that are not operated at their zero-expansion temperatures [46–48]. Beat note with active RAM control shows the best stability at averaging times >0.4 s. At longer times of >10 s, the increasing trend of the Allan deviation does not indicate the performance limitation of the RAM control but is caused by the drift of the two optical cavities.

Fig. 6
figure 6

Frequency stability of the optical heterodyne beat between two cavity-stabilized Nd:YAG lasers and RAM-induced frequency instabilities in the two systems. Solid circles are the contributions when the active control of RAM is turned on in both systems. The dashed horizontal line indicates the stability limit imposed by the thermal noise of the optical cavities

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The measurement noises, which include shot noise, electronic noise from photodetectors, and RAM-related noise, are potential barriers as the thermal noise of the optical cavity is further reduced. First of all, increasing the PDH discrimination slope by lowering the cavity loss and maximizing the mode-matching efficiency is a feasible and effective way to proportionally reduce the susceptibility of the system to various measurement noises. It is also helpful to tentatively evaluate the necessity of an active RAM control for ultra-stable optical oscillators aiming at stabilities of 10−16–10−17. For this evaluation, we consider the case of a 10-fold increase in the current discrimination slope and the RAM-induced frequency instability will be reduced accordingly to less than 1 × 10−16 up to 1,000 s, reaching to its minimum of 2 × 10−17 at 2 s, whereas for the case of only passive stabilization, the frequency instability is above 1 × 10−16 on timescales longer than 10 s (c.f. Figs. 3, 6).

4.2 Spectral linewidth

Apart from the noise spectrum and frequency stability, the linewidth is a commonly used parameter to evaluate the performance of optical oscillators. As those of all practical lasers, the frequency noise of highly stable optical oscillators is dominated by low-frequency components, resulting in a varying spectral linewidth with the duration of the measurement [72, 78]. According to a simplified model introduced by Di Domenico et al. [72], the linewidth of a sinusoidal signal modulated by a flicker frequency noise Sν = a/f α (1 ≤ α ≤ 2) is expressed as

$$\Delta \nu_{\text{FWHM}} = f_{m} \left( {\tfrac{8\ln 2}{\pi }} \right)\left[ {\tfrac{{\left( {f_{m} T_{0} } \right)^{\alpha - 1} - 1}}{\alpha - 1}} \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} ,$$

where α and a are the power law and the strength of the flicker noise, respectively, T 0 is the measurement time, and \(f_{m} = \left( {\tfrac{{a\pi^{2} }}{8\ln 2}} \right)^{1/2}\) is a characteristic frequency above which the noise components, while shaping the two wings of the spectral line, have a negligible contribution to the linewidth.

We experimentally verify this model by adding flicker frequency noises with different strengths to a RF oscillator [78] and also by using the two frequency-stabilized Nd:YAG lasers with their frequency noise predominantly coming from the thermal noise of the optical cavity. The flicker frequency noise (α = 1.5) is generated by an operational amplifier and then used to modulate the RF oscillator. The line profile of the modulated RF oscillator or the beat note of the two Nd:YAG lasers is obtained by averaging 10 center-aligned power spectra, each of which is a single-shot, time-dependent spectrogram [79] recorded by the FFT analyzer with a fixed measurement time.

Figure 7 shows the experimentally determined and calculated linewidths with varied measurement time as well as modified power law and strength of the flicker frequency noise. For the parameters we covered, the measurements agree with the calculations according to the simplified model. Note for the data located even in the hatched area (T · f m  < 5), where the model shall become less accurate [72], reasonable agreements still hold. Figure 8 shows six typical line profiles of the optical heterodyne beat between the two Nd:YAG lasers stabilized to ultra-stable optical cavities. In choosing between the Lorentzian and Gaussian profiles to fit the experimental spectra shown in the figure, the former provides better results, demonstrating the effect of a low-frequency cutoff introduced by the finite measurement time, which is 4.096 s in this case.

Fig. 7
figure 7

Linewidth versus measurement time. Symbols are experimental data and thin curves are calculations according to a simplified model [72]. The RF and optical carriers are modulated by flicker frequency noises whose power law α and strength a are indicated in the figure. Three data points at bottom (solid squares) are measured with the optical heterodyne beat of two Nd:YAG lasers. The rest are obtained with a RF oscillator. The characteristic experimental parameter T · f m = 5 is indicated by a thick straight line below which the theoretical predictions become less accurate [72]

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Fig. 8
figure 8

Spectral line profiles of the optical heterodyne beat between two Nd:YAG lasers stabilized to ultra-stable optical cavities. Each profile (symbols) is obtained by averaging 10 center-aligned spectra measured with a FFT analyzer using a Hanning window and a single-shot acquisition time of 4.096 s. Solid lines are fitted Lorentzian profiles to which the corresponding linewidths (FWHM) are attached. Except for the rightmost one, the line profiles are deliberately shifted in vertical for clarity

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5 Conclusions

Active control of RAM is implemented in two ultra-stable optical oscillators based on 1,064-nm Nd:YAG lasers that are locked to high-finesse optical cavities. Instead of being extracted in the PDH-frequency-locking path, RAM is detected in a different path and an added EO crystal inserted in front of the EOM improves the control bandwidth. The experimental verification of this control scheme and an accompanying theoretical analysis suggest that the effectiveness of the control is not affected by the optical cavity even when it is on resonance with the laser, provided that the birefringence of the EO crystal is the primary source of RAM.

The optical heterodyne beat of the two Nd:YAG lasers shows a frequency stability of 2 × 10−15 at averaging times from 0.2 to 10 s. With a measurement time of 4.096 s, the beat note exhibits a Lorentzian profile with 1-Hz linewidths (FWHM). Furthermore, measurements performed on a noise-modulated RF oscillator and two Nd:YAG lasers confirm a simplified model [72] that provides a straightforward connection of the parameter to the flicker frequency noise of the oscillators. With the active control, the instability of RAM is below 1 × 10−6 at averaging times of 0.02–1,000 s, reaching a minimum of 2 × 10−7 at ~2 s. A preliminary evaluation based on the current level of RAM and the optimized PDH-frequency-locking parameters suggests that for applications exploring 10−16–10−17 levels of frequency stability on timescales extending to 1,000 s, the impact of RAM and its active control will have to be considered.

The optical oscillators developed and analyzed here will be used to study the detailed properties of the thermal noise. Specifically, we plan to probe the spectral correlations of the thermal noises in two cavities, an intermediate step that is necessary for developing correlated measurement that may increase the sensitivities in a variety of laser interferometers.