1. Introduction

The Earth rotation rate, which is a key element of the Earth orientation parameters (EOP), has been measured with different gyroscopes [1–8]. Based on the Sagnac effect and laser technology, the optical method of Earth rotation measurement had made great progress in the past century [3,4]. The optical gyroscopes using chip-based structures, fiber rings or even giant fiber networks have been proposed and prove their potential in this field [5–8]. In the last two decades, benefited from the improvements of the mirror coating technology, the sensitivity of laser gyroscopes have been greatly enhanced, especially for large-scale ring laser gyroscopes (RLGs) that are used in Earth rotation sensing, which play an important role in the fields of rotational seismology, geodesy, and fundamental physics [9–16]. Among these excellent RLGs, the “G-ring” has been used successfully in the detection of the Chandler wobble and the annual wobble of Earth [14,17]. Another type of large-scale gyroscope is passive resonant gyroscopes (PRGs), that are operated by locking two external laser beams onto the eigen-modes of a high finesse optical ring cavity. With the development of laser locking techniques, the PRGs have also been considered for precision Earth rotation rate measurement [18–23].

Though operated in different forms, the PRGs and RLGs are based on the same principle. For a laser gyroscope which has a ring cavity, one can obtain the Sagnac frequency $f_\textrm {s}$ as [4]:

Here $\vec {\Omega }_\textrm {E}$ denotes the rotation vector of Earth and $\vec {K}=(4A\vec {n})/\lambda P$ is the scale factor of the gyroscope, where $A$ and $P$ are the enclosed area and the perimeter of the cavity while $\vec {n}$ denotes the normal vector of the area enclosed by the cavity, and $\lambda$ is the laser wavelength. For a traditional laser gyroscope made of a square ring cavity, the stability of the scale factor is free from the cavity length fluctuation to the first order, because the laser wavelength $\lambda$ would always compensate the changes of the perimeter $P$ and the enclosed area $A$ when the resonant conditions are maintained [22,24]. For this reason, the Sagnac frequency is insensitive to the small fluctuations of the cavity length when we only take the scale factor into account. On the other hand, since we use two injected laser beams to measure the frequency difference of two eigen-modes of the ring cavity, the fluctuation of the beat frequency contributes the majority of the noises in the rotation measurement [22].

One of the major limiting factors of laser gyroscopes is the backscattering effect. Due to imperfections of mirror surfaces, a small amount of laser light in the cavity is scattered back into the cavity eigen-mode in the other direction and amplified by the gain media in RLGs. This backscattering signal, which has the same frequency with the Sagnac signal, would experience random phase variations induced by position variations of the cavity mirrors. One of the solutions in RLGs is to actively control the mirrors’ positions by locking the perimeters or diagonals of the square ring cavities [17,25,26]. Correction of back-scatter-induced systematic errors can also solve this problem to some extent [27].

By locking two laser beams to the eigenmodes in the different directions of the ring cavity, the PRGs can also obtain the Sagnac signal from the beating of the two beams. Though there are no gain media in the PRGs, the backscattering signal disturbs the error signal in the laser locking process and contribute a frequency offset between the laser and the eigen-modes of the cavity. In order to solve this issue, the PRGs are preferably run in a split-mode scheme where the laser beams in the clockwise (CW) and the counter-clockwise (CCW) directions are in resonant with different longitudinal modes of the ring cavity [19,21]. In this way, the frequency difference of the two laser beams contains not only the Sagnac frequency, but also an integer number of the free-spectral-range (FSR) of the cavity. This split-mode operation pushes backscattering signals to much higher frequency beyond the bandwidth of locking process. Thus, the backscattering disturbance can be isolated in laser locking process. However, the effects of backscattering signal in detection process cannot be suppressed. And to make matters worse, the detected signal is now sensitive to the cavity length drift. The FSR variation induced by cavity length change thus becomes the dominant noise in the long-term operation of PRGs. This noise can be corrected in data processing by a digital filter which is obtained by monitoring the length fluctuation of the ring cavity [21]. In order to realize a real-time measurement, an active compensation scheme is necessary, which requires a sufficient locking bandwidth and a large dynamic range for such a long cavity length compensation. Meanwhile, for long-term operation of the gyroscope, it is necessary to stabilize the cavity to a length standard. Therefore, a complicated and costly ultra-stable laser system is utilized [23,28]. Another drawback of the split-mode operation is that the output of the laser gyroscope is the sum of the Sagnac frequency signal and the FSR signal, so that it is suitable for the variation measurement of the Earth rotation rate, but is inapplicable if one needs to measure an absolute value of it.

In this paper, we propose a phase-sensitive heterodyne detection scheme in the large-scale PRGs to avoid the above mentioned shortcomings of a split-mode operation gyroscope. In our scheme, the output of the laser is split in two and is injected into the CW and CCW directions simultaneously. We use the Pound-Drever-Hall (PDH) method to lock the beams to the cavity [18]. The laser frequencies of these two beams are arranged for the split-mode operation, with one FSR frequency difference. In addition to this type of traditional split-mode operation arrangement, we prepare another laser beam and inject it into the gyroscope in one of the directions, for instance, the CW direction. The additional laser frequency is adjusted to be two FSRs away from the original CW laser beam’s frequency. The beat signals between the CCW laser and the two CW lasers are detected by a balanced photo-detector (BPD). Through a self-demodulation process, we can obtain a two-fold increase of the Sagnac frequency. By using this method, both the backscattering noise and the FSR frequency fluctuation noise can be suppressed at the same time. Moreover, benefited from the doubled signal frequency, the equivalent scale factor is doubled without changing the structure or optical wavelength of the PRG. With the enhancement of the SNR, the quantum noise limit of PRGs has aslo been suppressed. Last but not least, we can obtain the FSR signal which can be used to establish a low-cost cavity length stabilization system with such a signal in the RF range.

2. Theoretical model of the phase-sensitive scheme

The proposed phase-sensitive heterodyne detection scheme is different from the scheme of normal common-mode or split-mode gyroscopes [29–33]. The schematic diagram is shown in Fig. 1. Four mirrors with the same ultra-high reflectivity form a ring cavity which acts as the core of the PRG. Two laser beams with frequencies of $\omega _1$ and $\omega _2$ are injected into the CW direction of the ring cavity and a third laser beam with a frequency of $\omega _3$ is injected into the CCW direction. A balanced heterodyne detection system is utilized to obtain the beat signals of the laser beams from two directions. The detection system includes a Mach-Zehnder interferometer (MZI) and a BPD. A self-demodulation system that contains an in-phase power splitter, a balanced frequency mixer and a band-pass filter is used to extract the desire Sagnac signal. Also, in some cases, we can obtain the high-quality signal directly from the radio-frequency (RF) sources of the AOMs, which can have higher intensity and better signal-to-noise ratio (SNR). The self-demodulation system thus can be easily replaced by a mixer. We record the Sagnac signal with a frequency counter.

 

Fig. 1. The schematic diagram of the phase-sensitive heterodyne detection in a large-scale passive resonant gyroscope. Three laser beams with frequencies of $\omega _1$, $\omega _2$ and $\omega _3$ are injected into a ring cavity. The transmitted optical fields in two directions, shown as $a_\textrm {CW}$ and $b_\textrm {CCW}$, are detected by a balanced heterodyne system. The beat signals in RF range are self-demodulated and recorded. BS: beam splitter, BPD: balanced photodetector, PS: power splitter, MIXER: frequency mixer, BPF: band-pass filter.

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The schematic diagram of the laser frequency distribution of the phase-sensitive heterodyne detection scheme is shown in Fig. 2. We set the angular frequency of one CW mode of the ring cavity as $\omega _0$, the frequencies of the two laser beams injected into the CW direction as $\omega _1=\omega _0+\omega _\textrm {F}$ and $\omega _2=\omega _0-\omega _\textrm {F}$, where $\omega _\textrm {F}=2\pi c/P$ is the angular frequency of the FSR. The frequency of the laser beam in the CCW direction is $\omega _3=\omega _0-\omega _\textrm {s}$, where $\omega _\textrm {s}=2\pi f_\textrm {s}$ is the angular Sagnac frequency. We assume the transmission fields with frequencies of $\omega _1$, $\omega _2$ and $\omega _3$ as $a_1 e^{i\omega _1 t}$, $a_2 e^{i\omega _2 t}$ and $b e^{i\omega _3 t}$, where $a_1$, $a_2$ and $b$ are the annihilation operators of these fields. To take the backscattering effect into consideration, we can use $\gamma _{21}$ and $\varphi _1$ to denote the amplitude and random phase of the backscattering light from the CCW direction into the CW direction, and $\gamma _{12}$ and $\varphi _2$ to denote the amplitude and random phase of the backscattering light from the CW direction into the CCW direction [27]. Moreover, the extra phase shift induced by the MZI for Sagnac signal measurement is also considered. The imbalance of the two arms of the MZI introduces a random phase noise, and is denoted as $e^{i\phi_{\textrm {MZI}}}$. Thus, the output optical fields of the cavity can be written with annihilation operators $a_\textrm {CW}$ and $b_\textrm {CCW}$ as

Notably, the center frequency $e^{i\omega _0 t}$ has the same contribution to each field and can be neglected by rotational-wave approximation. Thus, we can set $\omega _0=0$ in the following calculation. The optical fields after the beam splitter can be written as [29]

 

Fig. 2. The schematic diagram of the frequency distribution of the phase-sensitive heterodyne detection scheme in a large-scale passive resonate gyroscope.

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We can calculate the signal and noise respectively by linearizing of the operators: $a_1=\langle a_1\rangle +\xi _1$, $a_2=\langle a_2\rangle +\xi _2$, and $b=\langle b\rangle +\xi _3$, where $\xi _1$, $\xi _2$ and $\xi _3$ denote the fluctuations of each transmitted field. To calculate the signal, we focus on the mean field and ignore these fluctuations. We assume that the three transmitted fields have the same amplitudes $\langle a_1\rangle =\langle a_2\rangle =\langle b\rangle =\alpha$ for simplicity. The photon number on the first port of the BPD is then

The four components in Eq. (4) represent different signals:

Comparing Eq. (8) and Eq. (4), we can see that the average intensity and the beat signal from the CW direction are the same in the two ports of the BPD, while the beat signals between the laser beam in the CCW direction and the two laser beams in the CW direction have the opposite phase. The backscattering term in two the ports is slightly different

Assuming that the two ports of the BPD have the same quantum efficiency $\eta _\textrm {d}$, we can obtain the output voltage signal of the BPD

The signal can be thought as an amplitude modulated signal with the carrier frequency of $\omega _\textrm {F}$ and an amplitude modulation frequency of $\omega _\textrm {s}$. The modulation signal can be demodulated by a double balanced frequency mixer

The signal after the mixer contains a DC value and four components with frequencies of $2\omega _\textrm {F}$, $2\omega _\textrm {s}$ and $2(\omega _\textrm {F}\pm \omega _\textrm {s})$, respectively. The desire Sagnac signal can be extracted by a band-pass filter centered at $2\omega _\textrm {s}$ and recorded by a frequency counter as

Owning to this scheme, the dominant noise caused by the backscattering is isolated in the laser locking system with split-mode operation and filtered out in the detection process via the BPF. In addition, we can obtain the pure Sagnac signal without the FSR signal, for which the noise induced by the cavity length drift can be suppressed dramatically over the traditional split-mode operation. Compared to the traditional common-mode PRGs, the signal frequency is doubled, which means we can increase the scale factor of the PRGs without enlarging the cavity or changing the optical wavelength. This advantage gives a lower quantum noise limit of rotational sensitivity of the PRGs which will be discussed in the next section. Moreover, we can naturally obtain the FSR signal, which is in the RF range. This enables us to build a simple and low-cost system to stabilize the ring cavity length onto a microwave frequency standard instead of a complicated optical reference system [23]. The extra phase noise induced by MZI, which also appers in RLGs, is not suppressed with this scheme. But this noise can be suppressed to be lower than the quantum noise limit, and we discuss it in Sec. 3.4.

3. Noise analysis

Unlike the traditional split-mode operation in the PRGs, our new scheme not only suppress the backscattering noise but also avoid the disturbance of the FSR frequency. In addition, the obtained Sagnac frequency is doubled, and it is equivalent to double the scale factor without increasing the physical size of the PRGs. The enhancement of the scale factor can offer a lower noise level and increase the rotational sensitivity of the PRGs. In this section, we discuss the dominant noises in phase-sensitive scheme of a PRG. We first discuss the noise induced by ring cavity length drift, and then prove that the quantum noise floor of this scheme is decreased by $\sqrt{3/8}$ compared with the split-mode operation. The residual backscattering noise, the extra phase noise induced by MZI, the electronic noise of the BPD, and the laser noise are also been discussedMoreover, we provide a numerical simulation to compare the proposed scheme and the triditional split-mode scheme based on the data of a $3~\textrm{m}\times 3~\textrm{m}$ heterolithic PRG that is developed in our lab (named HUST-1) [23]. The corresponding Sagnac frequency of HUST-1 is around $104~\textrm{Hz}$. The comparison and the noise contributions are summarized in Fig. 3.

 

Fig. 3. Comparison of the phase-sensitive and the traditional split-mode scheme: Quantum limit of rotational sensitivity of PRGs in the phase-sensitive regime (red curve) and the traditional split-mode (blue curve). Rotational noise induced by cavity length drift of the phase-sensitive (pink curve) and the traditional split-mode (green curve) scheme. The original backscattering noise (brown dot line) and residual backscattering noise after common mode rejection (purple dot line). Rotational noise limited by the laser noise (gray line) and the extra noise of MZI (dark blue dashed line)

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3.1 Cavity length noise

Although the FSR signal has been isolated in phase-sensitive scheme, the cavity length fluctuation cannot be neglected in the PRGs. This is due to the finite servo-loop-gain in the laser locking process, in which the contribution is inversely proportional to the closed-loop gain [19,21–23,28]. We can calculate the relative Earth rotational noise induced by the cavity length fluctuation in traditional spit-mode PRGs and phase-sensitive heterodyne PRGs, respectively [28]

Here $\delta \Omega _\textrm {TSM}$ and $\delta \Omega _\textrm {PS}$ denote the rotational noise of the traditional split-mode scheme and the phase-sensitive scheme, respectively, $\delta {P}$ is the fluctuation of the cavity perimeter, $G_\textrm {p}$ is the closed loop gain, and $f_\textrm {laser}$ is the laser frequency. Compared to the traditional split-mode scheme, the phase-sensitive scheme is not only free from the fluctuation of the FSR frequency, but also has a two-fold normalized frequency. Thus, the rotational noise induced by the cavity length fluctuation can be dramatically suppressed via the phase-sensitive operation, and a remarkable enhanced performance of a PRG can be expected.

According to our measurement, we assume that the free cavity length noise spectrum of the PRG is $\delta P=10^{-9}/f~\textrm{m}/\sqrt{\textrm{Hz}}$ where $f$ denotes the frequency offset from the Sagnac signal, which acts as a random walk noise [23,28]. The loop gain of the laser locking is $G_\textrm {p}=10^{10}$ and the laser frequency is $f_\textrm {laser}\approx 281~\textrm {THz}$ [21,22]. The rotational noise induced by the cavity length drift in the traditional split-mode scheme and the phase-sensitive scheme can be obtained from Eq. (14) and (15), and are shown in Fig. 3 with green and pink dashed lines, respectively. Benefited from the immunity of the FSR frequency and the enhancement of the signal frequency, the rotational noise limited by the cavity length drift can be improved by at least 3 orders of magnitudes. In order to suppress the contribution of the cavity length noise down to below the quantum noise level, a feedback control system with nearly $\textrm{1}~kHz$ bandwidth is needed for the traditional split-mode PRG, while a bandwidth of $1~\textrm{Hz}$ is sufficient for the PRG in the phase-sensitive scheme. In addition, we can obtain the FSR signal directly through the phase-sensitive scheme of the PRG. With the cavity length signal in the RF range and much lower feedback bandwidth requirement, a low-cost and integrated cavity-length-stabilization system can be established.

3.2 Quantum noise limit

As described in Sec. 2., the Sagnac signal is obtained from the beating of transmitted fields of the ring cavity. Benefited from the ultra-narrow linewidth of the cavity (around 200 Hz for HUST-1), the classical noise of the optical field can be largely filtered out by the ring cavity. In addition, the backscattering noise of each transmitted field is suppressed by the common mode rejection of the BPD. The noise in this scheme is mainly due to the quantum fluctuation. Although the standard quantum limit of optical gyroscopes due to the radiation pressure backaction has been well studied, we still need to discuss the quantum imprecision noise in our proposed scheme [34]. In order to assess the quantum noise limit of a PRG in the phase-sensitive scheme, we ignore the backscattering terms and the phase shift induced by the MZI, and we add the quantum fluctuation to Eq. (3),

Note that the transmitted fields $a_1$, $a_2$ and $b$ are coherent state here, which obey

Since we only consider the noise around $\omega _\textrm {F}\pm \omega _\textrm {s}$, we define $A[x]_{\omega _i}$ as the amplitude of signal $x$ with a frequency of $\omega _i$. The output voltage noise of the BPD is

In an ideal case, the demodulation process does not introduce extra noise. Together with Eq. (10), the phase imprecision noise spectrum, which equals to voltage noise to signal ratio, takes the form [29]

The quantum phase imprecision noise limits the performance of laser gyroscopes in two ways. One is caused by SNR in the detection process. It acts as a white phase noise and is in proportional to the analyzed frequency in the form of frequency noise. The other noise is induced by frequency-phase transformation response of the ring cavity which acts as a white frequency noise. All of the frequency noises should be normalized to the Sagnac frequency to obtain the noise of the relative Earth rotational rate. The total shot noise limit in the phase-sensitive scheme can be written as

Here $f_\textrm {cav}$ denotes the linewidth of the ring cavity. In comparison with a traditional split-mode PRG, the frequency noise is $\sqrt{3/2}$ times larger. It seems that the phase-sensitive heterodyne detection process increases the noise. However, benefiting from the doubled equivalent scale factor which give a larger normalized signal frequency at $2f_\textrm {s}$, the relative noise level of the phase-sensitive scheme actually become lower. It is $\sqrt{3/8}$ times smaller compared with the traditional split-mode detection. Assuming the total power of the output beams of HUST-1 is as large as $0.1~\textrm{mW}$. According to Eqs. (21) and (22), we can calculate the rotational noise induced by the quantum noise in the traditional split-mode and the phase-sensitive PRG, as shown in Fig. 3 with blue and red curves, respectively. Benefited from the enhancement of signal frequency, we can reach a lower quantum limit of PRGs at a level of $10^{-12}~\textrm{rad}/\textrm{s}/\sqrt{\textrm{Hz}}$ without utilizing a complicated non-classical optical state.

3.3 Residual backscattering noise

In the phase-sensitive scheme, the Sagnac signal is in the phase quadrature, while the backscattering noise is in the amplitude quadrature. Thus, the backscattering noise can be suppressed by common mode rejection of the BPD. The BPDs have been widely applied in the experiments of quantum nonclassical state or quantum communications [35–38]. The two key elements about the BPD are the electronic noise and common mode rejection ratio (CMRR). In this section, we discuss the effect of the CMRR of the BPD in rotational measurement. The electronic noise of the BPD is discussed in Sec. 3.5. The finite CMRR of the BPD introduces the residual backscattering signal into the Sagnac signal. The output voltage of the BPD becomes

Then the signal detected by the frequency counter contains the Sagnac signal and the residual backscattering signal

We can obtain the maximum phase shift induced by the backscattering and its fluctuation

For simplicity, we have ignored the variation of $\phi _\textrm {MZI}$ here. Clearly, the maximum backscattering noise can be obtained when $\textrm{sin}(\varphi _2-\phi _\textrm {MZI})=1$, and $\textrm{sin}(\varphi _1+\phi _\textrm {MZI})=-1$,

Considering the fluctuation of the backscattering induced by the cavity length fluctuation $\delta \varphi _i=\delta P/\lambda$, the upper limit of the backscattering noise in rotational signal is

We assume that the backscattering signal in the two directions have the same intensity $\gamma _{21}=\gamma _{12}=0.05$, and the CMRR of the BPD as high as 75.2 dB [37]. The original backscattering noise is shown in Fig. 3 with brown dot line. The residual backscattering noise after common mode rejection of the BPD is also shown as purple dot line. As an amplitude noise, the backscattering signal is suppressed nearly 4 orders of magnitudes and is much lower than the quantum noise limit.

3.4 Extra phase noise induced by MZI

In order to measure the frequency difference of the output lasers of the ring cavity, an MZI is utilized. The air flow and the motion of the mirrors cause the imbalance of the two arms of the interferometer and introduce a random phase noise in the measurement process of the Sagnac signal [22]. This extra phase noise not only exists in PRG but also in RLGs or fiber-optic gyroscopes. We denote $\delta L_\textrm {MZI}$ as the random length imbalance noise of the MZI. The random phase noise induced by this interferometer and its contribution in the rotational noise can be obtained as

By replacing the heterolithic MZI with a temperature controlled integral prism beam combiner, the random phase noise induced by the interferometer can be dramatically suppressed. We assume that the random walk noise spectrum of the beam combiner is $\delta L_\textrm{MZI}=10^{-12}/f~\textrm{m}/\sqrt{\textrm{Hz}}$. The rotational noise induced by the interferometer can be obtained by Eq. (29) and is shown in Fig. 3 with a dark blue dashed line. The relative rotational noise $\delta \Omega _\textrm {MZI}/\Omega _\textrm {E}$ is around $9\times 10^{-9}~1/\sqrt{\textrm{Hz}}$, which is lower than the quantum noise limit.

3.5 Electronic noise of the BPD

In the quantum optical experiment, the noise of the BPD is very important. The electronic noise of the BPD mainly comes from the junction capacitance of the photodiodes, the voltage and current noise of the amplifiers, the thermal noise of the resistances, and the power supply. In phase-sensitive scheme, the SNR is dominated by the intensity and its fluctuation of the optical fields and the noise of the BPD. One of the advantages of the PRG is that the output laser power can be much higher than that in RLGs. The output power of the three laser beams in our scheme are 0.1 mW and the wavelength of the laser is 1064 nm as discussed in Sec. 3.2, and the corresponding shot noise is about $4.3~\textrm {pW}/\sqrt{\textrm{Hz}}$. The noise equivalent power (NEP) of some photo-detectors can be as low as $50~\textrm{fW}/\sqrt{\textrm{Hz}}$ (Excelitas Si APD LLAM-1060-R8BH), which is far less than the quantum noise limit and can be neglected.

3.6 Laser frequency noise

The laser frequency noise can also affect the performance of the PRGs. Though it can be suppressed by the laser locking process, this noise should not be ignored. The effect of laser frequency noise for a $1~\textrm{m}\times 1~\textrm{m}$ PRG has been well studied [22]. With the utilization of an Nd:YAG laser which has much lower noise and higher loop gain, we can assume the residual laser frequency noise can be as low as $10^{-6}~\textrm {Hz}/\sqrt{\textrm{Hz}}$. The rotational noise induced by laser frequency noise is shown in Fig. 3 with a gray line.

4. Discussion and conclusion

In conclusion, we propose a new method to realize PRGs. By injecting three separated laser beams in three adjacent longitudinal modes of the ring cavity in opposite directions, the performance can be improved significantly over the traditional split-mode PRGs. The noise induced by the cavity length fluctuation can be dramatically suppressed below the quantum noise limit where only a narrow band cavity-length-stabilization system is needed. Compared to the common-mode PRGs, the PRGs with this scheme is not disturbed by the backscattering effect owing to the split-mode operation and common mode rejection of the BPD. Moreover, we can obtain a two-fold Sagnac signal which increases the equivalent scale factor of the laser gyroscope without change the physical size or the optical wavelength. In another word, the phase-sensitive scheme gives an opportunity to reach a lower quantum noise limit of PRGs. Another advantage of this design is that we can obtain the FSR signal directly, which can act as the cavity length signal in the RF range. Thus, the stabilization of the cavity length can be achieved by a low-cost and compact system which is important for navigation gyroscopes. This scheme certainly increases the complexity of the PRG system. As one more beam is injected into the cavity, more servo systems are needed. Also, how to avoid the interaction between the three lasers, especially the two laser beams in the same injection direction may be a technical challenge. Nevertheless, once successfully implemented, a phase-sensitive heterodyne detection PRGs will have a much better performance, profiting from the suppression of cavity length noise and backscattering noise, and also the enhancement of the detected SNR. Moreover, the principle of resonant fiber-optic gyroscopes and RLGs are the same as that of the PRGs in that they all measure the rotational rate by the Sagnac signal with a ring cavity and an interferometer. Clearly, the phase-sensitive heterodyne scheme is also applicable for these optical gyroscopes. With the dramatically improvement of the rotational sensitivity, the measurement of the length of day or even the test of the General Relativity with optical gyroscopes can be realized.