Laser frequency noise induced error in resonant fiber optic gyro due to an intermodulation effect

Huilian Ma; Yuchao Yan; Linglan Wang; Xu Chang; Zhonghe Jin

doi:10.1364/OE.23.025474

1. Introduction

The resonant fiber optic gyro (RFOG), based on the Sagnac effect as the traditional interferometric fiber optic gyro (IFOG), was first proposed by Ezekiel and Balsamo on the basis of the study of the passive resonator laser gyro [1]. The basic theory of the Sagnac effect induced by the rotation enhanced by a passive optic ring resonator is the multi-beam interference, which makes the RFOG achieve the same precision with a shorter fiber length, a smaller volume and a lower weight, as well as fit the need of miniaturization.

In the RFOG, the rotation rate readout is linearly proportional to the resonant frequency difference between the clockwise (CW) and counterclockwise (CCW) lightwaves propagating in an optical fiber ring resonator (OFRR) through the Sagnac effect. Perhaps the greatest technical challenge for the RFOG developers has been the demand for precise determination of the CW and CCW resonant frequencies [2]. A narrow-linewidth laser is one of the key optical components in the RFOG. The influence of the laser linewidth on the theoretical shot noise limited sensitivity is more considered in previous studies. A bias drift in the range of 0.1 to 0.3 deg/h has been predicted for an resonant micro optic gyro (RMOG) with a waveguide-type ring resonator based on the investigation of the influence of the laser linewidth on the theoretical shot noise limited sensitivity [3]. An online laser frequency noise observation method is also proposed [4]. It is markedly deteriorated when the laser linewidth and the resonator spectral linewidth are in the same magnitude. However, the influence on the sensitivity can almost be ignored with a laser linewidth one order smaller than the resonator spectral linewidth.

Nevertheless, some practical experiment results indicate that the effect on the performance of the RFOG exerted by the laser seems to be more predominant than the shot noise limited sensitivity deterioration related to the linewidth. Imai et al. attained different sensitivities of 0.36°/h and 3.6°/h by applying two laser sources, a YAG laser with a linewidth of 5 kHz and a semiconductor laser with a linewidth of 30 kHz, respectively on one fiber resonator system [5]. Although the spectral linewidths of the two laser sources are both much smaller than the intrinsic spectral linewidth of the resonator, the corresponding sensitivity is of one order of magnitude different than that of another. Strandjord and Sanders achieved a precision of 0.1°/h by the utilization of a YAG laser as the laser source of an RFOG. This precision is two orders of magnitude higher than the shot noise limited sensitivity. They indicated that the wideband laser frequency noise greatly affected the random performance of the RFOG [6–8]. Actually, as early as 1977, the significance of the laser frequency noise was appreciated by Ezekiel and Balsamo at the inception of the passive resonator gyro [1]. Thus, the effect of the laser frequency noise on the resonant gyros still needs further research.

The nonlinear intermodulation effect is well known in the field of communication and microwave system. An intermodulation effect arises when multiple signals pass through a nonlinear system. The laser frequency noise around even multiples of the modulation frequency f_M sets a limit to the laser frequency stability when the laser was locked to an atomic resonator [9, 10]. This effect has been introduced to the resonator optic gyro by Ma et al. in 2010 [11]. It is related to the nonlinear behavior of the optical ring resonator, to the necessity of modulation and demodulation process used for the resonance-center detection. Laser noise within a frequency region close to even harmonics of the modulation frequency 2nf_M (n is a positive integer) was translated down to baseband via the modulation and detection process. Qiu et al. also showed that the laser noise at high frequencies was deleterious to gyro performance and proposed a solution based on laser stabilization and filtering [12].

The purpose of this work is to derive a quantitative value of the effects of laser frequency noise on the performance of sinusoidal phase modulation RFOGs due to the nonlinear intermodulation effect. An intermodulation calculation model based on the OFRR is obtained and the intermodulation errors introduced by two types of narrow-linewidth lasers are analyzed. In order to experimentally investigate this effect, an additional phase modulator (PM) is added to the RFOG. Then the laser frequency noise is simulated by phase modulating this additional PM at different frequencies. Large fluctuations have been observed at the gyro output with modulating signals at 2f_M and 4f_M, respectively. It is almost the same order of the initial output fluctuation with modulating signal at 6f_M. The open loop gyro output keeps unchanged while applying modulating signals at other frequencies. The aforementioned results are in agreement with theoretical expectations. In addition, different detection precisions are attained by using three different laser sources respectively in the same RFOG.

2. Principle and simulation

2.1 Laser frequency noise

We assume that the amplitude noise is negligible, and that the spectral linewidth is caused mainly by the frequency noise. In such a case, the electric field of a single longitudinal mode laser is expressed as

E (t) = \sqrt{P_{0}} \cos (2 π v_{0} t + φ (t))

where P₀ is the average optical power, ν₀ the mean optical frequency and φ(t) is the phase noise. The single-sided frequency noise spectrum can be modeled at frequencies below the relaxation oscillation frequency, as power-dependent white noise and power-independent 1/f noise [13]

S_{F} (f) = C + \frac{K}{f} f \geq 0

where C and K are the white and the 1/f noise components, in units of Hz²/Hz and Hz³/Hz, respectively. When the frequency noise spectrum is white, the laser output spectrum is a Lorentzian lineshape and the 3dB linewidth Δf_L is approximately given as [12]

Δ f_{L} = π C = π Δ v_{r m s}^{2}

where Δυ_rms is the power spectral density of the white frequency noise in units of Hz/Hz^1/2. As the 1/f noise becomes predominant, the laser output spectrum becomes more like a Gaussian distribution.

Various lasers have been demonstrated with a narrow-linewidth and a wide frequency-tuning range. However, their frequency noise spectrums are different from each other even when they have equal linewidths. In the case of semiconductor lasers, the dominant contribution to the linewidth is the white noise. Unlike semiconductor lasers, fiber lasers and YAG lasers are not dominated by the white noise. Instead, their noise characteristics more closely resemble a low-pass filter function. The linewidth due to the white noise can be as low as 1 Hz in the YAG laser with a kHz-level measured linewidth [14]. A typical spectral linewidth of a highly-coherent fiber laser ranges from 1 kHz to 100 kHz while the white noise contributed linewidth is 100-1000 Hz [15]. The following analysis demonstrates that the intermodulation effect due to the laser frequency noise mainly results from the white noise.

2.2 Principle of the intermodulation effect

In resonant optical gyros, the operation of the primary resonant frequency feedback loop is based on the first harmonic lock-in technique. The laser beam before being launched into the resonator is phase modulated at a frequency of f_M. The demodulated output of the lock-in amplifier (LIA) is low-pass filtered by a frequency feedback loop with a bandwidth of B. B is generally much smaller than f_M. Therefore, only the new frequency spectrum component mixed by the narrow-linewidth laser frequency noise and the modulation frequency can be detected within the signal bandwidth. For instance, the frequency component between f_M-B to f_M + B is mixed by the laser frequency noise in a frequency band from 2f_M-B to 2f_M + B and the phase modulation frequency f_M, and it is detected by the LIA as the demodulated signal. We thus introduce the following Rice representation of the laser frequency noise in a set of narrow spectral windows around f_N [16]

Δ v (t) = P (t) \sin (2 π f_{N} t) + Q (t) \cos (2 π f_{N} t)

where P(t) and Q(t) are slowly varying amplitudes. Assuming that Δν(t) is the white noise in the very narrow bands around f_N, then P(t) and Q(t) are not correlated and their one-sided power spectral densities (PSDs) have the following property

S_{P} (f \leq B) = S_{Q} (f \leq B) = 2 S_{F} (f = 2 f_{N})

where S_F is the PSD of the laser frequency noise in units of Hz²/Hz.

Due to Eq. (5) of Rice distribution, the narrow band frequency noise can be regarded as an equivalent sinusoidal phase fluctuation, which can be further equivalent to the sinusoidal phase modulation. Thus, the narrow band frequency noise at the frequency of f_N can be regarded as an equivalent phase modulation with a modulation frequency of f_N and a modulation coefficient of M_N written as

M_{N} = \sqrt{2 B \cdot S_{P} (f_{N})} / (2 π f_{N}) = \sqrt{B \cdot S_{F} (f_{N})} / (π f_{N})

With the laser frequency noise equivalent to the phase modulation, the output field of a laser with a narrow band frequency noise at f_N can be expressed as

E_{L a s e r} (t) = \sqrt{P_{0}} \exp j [2 π ν_{0} t + M_{N} \sin (2 π f_{N} t)]

where f_N should include every possible value for the frequency noise of a practical laser ranges over a wide frequency band. The following analysis comprehensively calculates the corresponding intermodulation error generated by the laser frequency noise under different values of f_N.

The lightwave propagated along one certain direction through the OFRR is taken into consideration. The calculation methods of the CW intermodulation noise and the CCW one are similar. The laser output field after the PM is expressed as

E_{P M - o u t} (t) = \sqrt{P_{0}} \exp j [2 π ν_{0} t + M_{N} \sin (2 π f_{N} t) + M_{1} \sin (2 π f_{1} t)]

Where M₁ is the modulation coefficient of the CCW lightwave, and f₁ is the modulation frequency.

Be analogous to the derivation of the double phase modulation process [17], the output field of the CCW lightwave at the photodetector (PD) is given as

\begin{array}{l} V_{PD - o u t} = G P_{0} \sum_{n = - \infty}^{\infty} \sum_{m = - \infty}^{\infty} \sum_{n' = - \infty}^{\infty} \sum_{m' = - \infty}^{\infty} J_{n} (M_{1}) J_{m} (M_{N}) J_{n'} (M_{1}) J_{m'} (M_{N}) \\ \cdot \exp j 2 π [(n - n') f_{1} t + (m - m') f_{N} t] h_{(m, n)} h_{(m', n')} \exp j (ϕ_{(m, n)} - ϕ_{(m', n')}) \end{array}

where G is the gain-related coefficient of the CCW lightwave, φ_(m,n) is the phase delay of the light field induced by the OFRR, and h_(m,n) is the transmission factor of the light field in the OFRR.

Equation (9) expresses the output voltage at the PD in consideration of the laser frequency noise, which is also the input voltage signal to the LIA and consists of complicated frequency components. In the phase modulation and synchronous demodulation technique, the LIA only detects the first harmonic component at the frequency of f₁ of the PD output. The LIA is only able to detect the voltage component satisfying the following condition

(n - n') f_{1} + (m - m') f_{N} = \pm f_{1}

For the values of n, n′, m and m′ are integers in full range, there is an infinity of the f_N to f₁ ratio. The first harmonic component in Eq. (10) consists of both the interference component between the modulation sideband, which is the to-be-detected signal, and the interference component between the modulation sideband and the laser frequency noise, which is the intermodulation error.

The size of each harmonic of the LIA input signal is correlated to the Bessel function coefficient in Eq. (9). The values of these Bessel coefficients will be analyzed and in the following calculation of the intermodulation error, the harmonic components with the Bessel coefficients of very small amplitudes will be ignored.

Due to the property of the Bessel function, when the modulation coefficient is smaller than 1, the Bessel functions of different orders satisfy

J_{0} (M_{N}) \approx 1, J_{1} (M_{N}) \approx M_{N} / 2, J_{1} (M_{N}) ≫ J_{2} (M_{N}) if M_{N} < 1

In an RFOG, the modulation coefficient is set to be 2.405 in order to suppress the backscattering noise. When the modulation coefficient is 2.405, except the zero-order Bessel function, the Bessel coefficients of the other orders decrease with the increasing order. The Bessel coefficient of 5-order is decreased by about 30 times compared to the first-order. Thus, when the modulation coefficient M₁ = 2.405, we just require calculation with of the condition of Eq. (9) satisfied and the order smaller than or equal to 4.

For a semiconductor laser with a linewidth of 100 kHz, the PSD of the white frequency noise Δυ_rms is approximately 178 Hz/ Hz^1/2 calculated from Eq. (3). For the frequency noise at a Fourier frequency greater than 100 kHz, the equivalent modulation coefficient M_N is 0.00356 according to Eq. (6), which is far smaller than 1. According to Eq. (13), we only take into account only when the order is −1, 0, or 1. The specific calculation of the intermodulation noise brought about by the laser frequency noise at the LIA output will be conducted on the basis of the phase modulation and demodulation process.

2. 3 Analysis and simulation of the intermodulation effect

The term about m in Eq. (10) is inexistent while the effect of the laser frequency noise is completely ignored. Hence, Eq. (10) is simplified as n-n′ = ± 1, which expresses the general phase modulation and the first harmonic demodulation process [18, 19].

In consideration of the existence of the laser frequency noise but ignoring the influence of the interference terms between the different laser frequency noise components, the intermodulation noise term m-m' is 0, and Eq. (9) can be simplified as

V_{L I A - i n} = 2 G P_{0} \sum_{n = - \infty}^{\infty} \sum_{m = - \infty}^{\infty} J_{m}^{2} (M_{N}) J_{n} (M_{1}) J_{n + 1} (M_{1}) h_{(m, n)} h_{(m, n + 1)} \cos [ω_{1} t + (ϕ_{(m, n + 1)} - ϕ_{(m, n)})]

where ω₁ is the angular modulation frequency. Compared with the expression of the demodulation curve without regard to the laser frequency noise [19], Eq. (12) manifests the fluctuation about M_N fully in the demodulation output, which means the laser frequency fluctuation is directly transformed to amplitude fluctuation of the LIA demodulation output, and the technique of the resonant frequency servo loop is necessary to suppress the fluctuation [20, 21].

When m-m′≠0, the signal at the frequency of f₁ is interfered by the laser frequency noise at twice the modulation frequency (2f₁), which can hardly be distinguished from the desired signal at f₁ containing the rotation information of the RFOG, and introduces the detection error at the LIA output which is unable to be reduced or suppressed by the resonant frequency servo loop. This is the to-be-studied intermodulation noise in the phase modulation and demodulation system introduced by the laser frequency noise. Reference [9] gives out the process of the formation of the intermodulation error at the gyro output induced by the laser noise at twice the modulation frequency in the gyro bandwidth after modulation and demodulation.

Theoretically, the intermodulation error generated by the laser frequency noise can be calculated by synchronously demodulating the input signal of the LIA expressed by Eq. (12) by the signal at f₁. However, the frequency components in Eq. (12) are too complicated to be computed, so the Bessel coefficients contributing less to the intermodulation error will be ignored.

In case of the modulation coefficient M₁ = 2.405 and in ignorance of the terms of high orders, the value of m-m' is ± 1 or ± 2, and the value of n-n' are the integers ranging from −8 to 8 due to the Bessel function feature. By Eq. (10), (m-m′) × f_N should be a non-zero multiple of the modulation frequency f₁, which tells that f₁ and f_N must satisfy

f_{N} = k \cdot f_{1} or f_{N} = k / 2 \cdot f_{1}

where k is a positive integer. The every possible value of f_N in Eq. (13) will be analyzed. All the conditions under which (m-m′) × f_N and (n-n′) × f₁ are able to be mixed and generate the frequency component at f₁ are listed and computed. Assuming that the RFOG is locked, in which case the laser central frequency is locked to the valley of the resonant curve, the amplitude transfer function of the curve is an even function, and the phase transfer function is an odd function.

When f_N = (2q-1)f₁ (q is a positive integer), the LIA demodulation output is 0, which means the laser frequency noise at odd multiple of the modulation frequency f₁ does not introduce the intermodulation noise.

When f_N = f₁/2, the LIA demodulation output is expressed as

\begin{array}{l} V_{o u t - 1 / 2} = - 2 G P_{0} J_{1}^{2} (M_{N}) {2 J_{4}^{2} (M_{1}) h_{3.5} h_{4.5} \cos (ϕ_{3.5} - ϕ_{4.5}) \\ + 2 J_{3}^{2} (M_{1}) h_{2.5} h_{3.5} \cos (ϕ_{2.5} - ϕ_{3.5}) + 2 J_{2}^{2} (M_{1}) h_{1.5} h_{2.5} \cos (ϕ_{1.5} - ϕ_{2.5}) \\ + J_{1}^{2} (M_{1}) [2 h_{0.5} h_{1.5} \cos (ϕ_{0.5} - ϕ_{1.5}) - h_{0.5} h_{- 0, 5} \cos (ϕ_{0.5} - ϕ_{- 0.5})] \\ + 2 J_{4} (M_{1}) J_{2} (M_{1}) h_{3.5} h_{2.5} \cos (ϕ_{3.5} - ϕ_{2.5}) \\ + 2 J_{3} (M_{1}) J_{1} (M_{1}) h_{1.5} h_{2.5} \cos (ϕ_{1.5} - ϕ_{2.5})} \end{array}

When f_N = 2f₁, the LIA demodulation output is expressed as

\begin{array}{l} V_{o u t - 2} = 4 G P_{0} J_{0} (M_{N}) J_{1} (M_{N}) \\ {J_{1} (M_{1}) J_{2} (M_{1}) [h_{2} h_{3} \cos (ϕ_{2} - ϕ_{3}) - h_{2} h_{1} \cos (ϕ_{2} - ϕ_{1})] \\ + [J_{1} (M_{1}) J_{4} (M_{1}) + J_{3} (M_{1}) J_{2} (M_{1})] [h_{3} h_{4} \cos (ϕ_{3} - ϕ_{4}) - h_{2} h_{1} \cos (ϕ_{2} - ϕ_{1})] \\ + J_{4} (M_{1}) J_{3} (M_{1}) [h_{4} h_{5} \cos (ϕ_{4} - ϕ_{5}) - h_{2} h_{3} \cos (ϕ_{3} - ϕ_{2})]} \end{array}

When f_N = 4f₁, the LIA demodulation output is expressed as

\begin{array}{l} V_{o u t - 4} = 4 G P_{0} J_{1} (M_{N}) J_{0} (M_{N}) \\ \cdot {J_{1} (M_{1}) J_{4} (M_{1}) [h_{4} h_{5} \cos (ϕ_{4} - ϕ_{5}) - h_{4} h_{3} \cos (ϕ_{4} - ϕ_{3})] \\ + J_{2} (M_{1}) [J_{3} (M_{1}) - J_{1} (M_{1})] [h_{2} h_{3} \cos (ϕ_{3} - ϕ_{2}) - h_{1} h_{2} \cos (ϕ_{2} - ϕ_{1})]} \end{array}

When f_N = 6f₁, the LIA demodulation output is expressed as

\begin{array}{l} V_{o u t_6} = 4 G P_{0} J_{1} (M_{N}) J_{0} (M_{N}) \\ {J_{3} (M_{1}) J_{4} (M_{1}) [h_{4} h_{3} \cos (ϕ_{4} - ϕ_{3}) - h_{2} h_{3} \cos (ϕ_{2} - ϕ_{3})] \\ + J_{4} (M_{1}) J_{1} (M_{1}) [h_{5} h_{4} \cos (ϕ_{5} - ϕ_{4}) - h_{1} h_{2} \cos (ϕ_{1} - ϕ_{2})] \\ + J_{2} (M_{1}) J_{3} (M_{1}) [h_{3} h_{4} \cos (ϕ_{3} - ϕ_{4}) - h_{3} h_{2} \cos (ϕ_{3} - ϕ_{2})]} \end{array}

When f_N>6f₁, its contribution to the total intermodulation error can be ignored thanks to the very weak laser frequency noise.

The above analysis demonstrates that when the system is locked, the laser frequency noise at the odd multiple of the modulation frequency does not output demodulation signal, thus it does not produce the intermodulation error. The laser frequency noise at the even multiple of the modulation frequency is demodulated and output, which generates the intermodulation error.

In comparison of Eqs. (14) and (15), the coefficient of Eq. (14) is proportional to J₁²(M), and the one of Eq. (15) is proportional to J₁(M_N)J₀(M_N). Due to Eq. (11), J₀(M_N)≈1 and J₁(M_N) = M_N/2 when M_N<1. The intermodulation noise generated by f_N = f₁/2 is considered to be two orders of magnitude weaker than the one generated by f_N = 2f₁ since M_N is smaller than 0.01 in the actual situation, therefore, it can be ignored in the following simulation.

The above theoretical deduction process is simplified and takes into account four terms only when the Bessel function order n equals to 1, 2, 3 or 4. Figure 1 calculates the demodulation output under three circumstances, which shows that compared with the demodulation outputs when n = 1, 2, 3, 4, the output when n = 5, 6 is an infinitesimal quantity. Therefore, the contribution of the Bessel functions of high orders to the intermodulation error is negligible in the calculation of the actual intermodulation noise.

Fig. 1 Demodulation output of the intermodulation error in consideration of the different Bessel orders.

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The above analysis demonstrates that the introduced intermodulation error varies by the laser frequency noise at different frequencies. The main source of the intermodulation error is the laser frequency noise at the even multiples of the modulation frequency, namely, f_N = 2nf₁ (n is a positive integer), especially for the frequency noise at twice the modulation frequency. The computing result is expressed as Eq. (15). Due to Eqs. (6) and (11), in ignorance of the effect of the high order terms of the Bessel function, it is obtained that when the modulation coefficient is 2.405, the intermodulation error introduced by the laser frequency noise at twice the modulation frequency at the LIA output is simplified as

\begin{array}{l} V_{0} = 2 G P_{0} \sqrt{2 B} \cdot S_{F} (2 f_{1}) / 2 f_{1} / 2 \\ \cdot [J_{4} J_{3} (V_{4, 5} - V_{3, 2}) + J_{1} J_{2} (V_{2, 3} - V_{2, 1}) + (J_{2} J_{3} + J_{4} J_{1}) (V_{3, 4} - V_{2, 1})] \end{array}

Where V_i,j = h_ih_jcos(ϕ_i-ϕ_j), and J_n = J_n(M₁).

Equation (18) points out that the demodulation output is proportional to the laser frequency noise at twice the modulation frequency and it increases with the decreasing of the modulation frequency. However, the maximum of the modulation frequency is tending to be affected by the slope of the demodulation curve. Generally, in order to achieve the maximization of the demodulation slope, an optimal modulation frequency related to the full width at half maximum (FWHM) of the resonator exists. Thus, in a real RFOG system, the choice of the modulation frequency should be in consideration of the both factors.

The open loop output voltage brought about by the laser frequency noise should be transformed to the angular velocity through the scale factor, which stands for the size of the signal. For the same rotation signal, the bigger the demodulation curve slope, and the higher the open loop output voltage is. Therefore, the equivalent frequency difference induced by the intermodulation frequency error is obtained by dividing the intermodulation voltage error by the demodulation curve slope. Due to the relationship between the frequency difference and the rotation rate, the rotation detection error induced by the intermodulation effect is expressed as

Ω_{I E} = \frac{\sqrt{2} G P_{0} S_{F} (2 f_{1})}{2 f_{1} \cdot k_{S F}} \cdot [J_{4} J_{3} (V_{4, 5} - V_{3, 2}) + J_{1} J_{2} (V_{2, 3} - V_{2, 1}) + (J_{2} J_{3} + J_{4} J_{1}) (V_{3, 4} - V_{2, 1})]

where k_SF is the scale factor

It is worth noting that Eq. (18) is the intermodulation error induced by the lightwave along one direction of the resonator. If the source of the intermodulation error is the white noise, since the RFOG detects the frequency difference between the CW and CCW lightwaves propagating in the OFRR, the total intermodulation noise should be multiplied by $\sqrt{2}$ . If the source is the 1/f noise, for the modulation frequencies of the two lightwaves are very close and the 1/f noise is reciprocal, the two parts of the intermodulation error generated by the CW lightwave and the CCW lightwave are likely to be canceled each other out. Equation (19) directly shows that the intermodulation error induced by the laser frequency noise is proportional to the PSD of the laser frequency noise. Due to the coefficients of the expression, the intermodulation error is inversely proportional to the modulation frequency. But on account of the influence on the terms within the brackets by the modulation frequency, the final relationship between the intermodulation noise and the modulation frequency needs further calculation.

3. Simulation and analysis

The intermodulation noise generated by different types of lasers will be calculated and analyzed. In an RFOG system with a total fiber length of 14 m, a diameter of 0.12 m, and a finesse of 17.8, the intrinsic spectral linewidth is 364 kHz. Consider the following two types of narrow-linewidth lasers:

(1) FL1 is a fiber laser from NKT Photonics with a linewidth of about 1 kHz. Assuming that the corresponding linewidth of the white noise is approximately 100 Hz, the effective amplitude of the white noise in the laser frequency noise PSD is calculated to be 5.6Hz/Hz^1/2.
(2) LD1 is a semiconductor lasers from RIO Inc. Its linewidth is about 3.5 kHz, which is mainly from the white noise. According to Eq. (3), the white noise in the PSD of the laser frequency noise has an effective amplitude of 33.4Hz/Hz^1/2.

Assuming that the corresponding linewidths of the white noise are approximately 0.1 kHz in the fiber laser FL1 and 3.5 kHz in the semiconductor laser LD1, respectively, when the modulation coefficient M₁ is chosen to be 2.405, the intermodulation error due to the laser frequency noise at twice the modulation frequency introduced by the lightwave of the single direction in the RFOG is shown as Fig. 2.

Fig. 2 Calculated intermodulation errors induced by a fiber laser FL1 and semiconductor LD.

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In Fig. 2, the intermodulation error is expressed as an effective value and the system bandwidth is 1 Hz. The rotation rate error caused by the intermodulation effect of FL1 and LD1 is always bigger than 0.002°/s, which is one order of magnitude higher than the limited sensitivity of the RFOG system. As seen from Fig. 2, the intermodulation error decreases by increasing the modulation frequency, and due to the relatively low white noise of FL1, its rotation rate error is always smaller than that affected by LD1.With a modulation frequency of 100 kHz, the intermodulation effect caused by the white noise of the FL1 and LD1 leads to a rotation rate error of 0.02°/s and 0.05 °/s. Since the RFOG detects the frequency difference between the CW and CCW lightwaves propagating in the OFRR, the total intermodulation error should be multiplied by $\sqrt{2}$ . Generally, the bias stability of the RFOG is 5 to 10 times smaller than the actual voltage fluctuation. Therefore, with the application of semiconductor laser LD1 with a white noise linewidth of 3.5 kHz, the bias stability limited by the intermodulation noise in the RFOG is still as high as 25~50°/h. The above theoretical analysis result will be further verified in the following experiments.

4. Experiments and results

According to the aforementioned analysis, when the resonant curve of the resonator is perfectly symmetric, only the laser frequency noise at twice the modulation frequency (2f_M) makes contribution to the intermodulation error. The intermodulation error is to be experimentally tested.

4.1 Simulation of the laser frequency noise by sinusoidal phase modulation

The schematic of the system is shown in Fig. 3. The polarization-maintaining fiber transmission-type OFRR with twin 90° polarization-axis rotated splices is the key rotation sensing element in the RFOG. For the further decrease of the polarization error, two in-line polarizers P_x and P_y are inserted [22, 23]. The cavity length is 14 m, and the diameter is 12 cm. The finesse and linewidth of the fiber ring resonator are 17.8 and 830 kHz, respectively. The LiNbO₃ phase modulators PM1 and PM2 are used to modulate the phase of the lightwave from the laser. The half-wave voltage of PM1 and PM2 are 3.78 V and 3.65 V, respectively. PM1 and PM2 are driven by the sinusoidal waves at different modulation frequencies to attenuate the backscattering noise. The amplitudes of the sinusoidal waves are 2.89 V and 2.79 V, respectively, carefully optimized to reduce the carrier [17, 18]. The achieved carrier suppression is more than 40 dB in each PM. The modulation frequency of the open-loop is f₁, and that of the closed-loop is f₂. In comparison with the general RFOG, our system adds a PM3 after the laser. According to Eq. (6) and Eq. (7), sinusoidal signals with different frequencies (f_N) and different modulation coefficients (M_N) are applied to PM3 to simulate laser frequency noise in the narrow band at different frequencies. The half-wave voltage of PM3 is 2.78 V. The CW and the CCW lightwaves from the resonator are detected by the InGaAs PIN photodetectors, PD1 and PD2. The output of PD2 is fed back through the lock-in amplifier LIA2 to the servo controller PI2 to reduce the reciprocal noises in the RFOG. The output of PD1 is demodulated by LIA1, and the gyro output is given out after a low-pass filter (LPF).

Fig. 3 Schematic diagram of the simulation of the laser frequency noise by sinusoidal phase modulation.

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The experiment parameters are set as follows. The modulation frequency of the open-loop is 105.15 kHz, the modulation frequency of the closed-loop is 103.15 kHz, the system bandwidth is 0.16 Hz (the integration time is 1 s), and the modulation voltage is 30 mV with a corresponding phase modulation coefficient of 0.0339. With the application of sinusoidal phase modulation to PM3, the intensity of the laser frequency spectrum has been strengthened, which makes the intermodulation effect be easily observed.

Figure 4 reflects the open-loop output fluctuation of the RFOG while applying signals with different frequencies to PM3. A fluctuation of as high as 4132 °/h is observed in the open-loop output as f₃ = 2f₁. The fluctuation decreases to 586°/h as f₃ = 4 f₁.When the modulation frequency of PM3 is six times of f₁, the fluctuation amplitude is almost the same order as the initial RFOG output. The result demonstrates that when the modulation frequency of PM3 is set to be the even multiple of the open-loop modulation frequency f₁ sinusoid-like fluctuation as high as thousands degree per hour appears in the gyro output, whose frequency varies with the modulation frequency of PM3. The open loop gyro output and the lock-in accuracy of the closed-loop keep unchanged while applying modulating signals with other frequencies. Only when the modulation frequency of PM3 is set to be the even multiple, as 2, 4 or 6 times, of the open-loop modulation frequency f₁, large fluctuation can be observed in the open loop output, as shown in Fig. 4.

Fig. 4 Open loop gyro output with the application of sinusoidal phase modulation.

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The further test indicates that with the modulation frequency of PM3 set to be the even multiple of the closed-loop modulation frequency f₂, the output phenomenon is similar with that of when the modulation frequency of PM3 is set to be the even multiple of the open-loop modulation frequency. The different intensity of the intermodulation error changing with the modulation frequency f₂, when the even multiple of the closed-loop modulation frequency is applied to the PM3, is shown in the Fig. 5. As the modulation frequency of PM3 is set to be other frequencies, the open loop output of the system remains unaffected.

Fig. 5 Closed-loop gyro output with the application even multiple of sinusoidal modulation frequency f₂.

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4.2 Application of the lasers with different linewidths to the practical RFOG

The schematic diagram of the RFOG is given out in Fig. 6. The PM3 which is applied to simulate the frequency noise of the laser in narrow band at different frequencies in the Fig. 3 is removed. The bias stability of the RFOG is tested with the application of different lasers with different linewidths. The details of FL1 and LD1 have been given out in section 3. LD2 is a semiconductor laser with a linewidth of about 100 kHz. The bias stability of the RFOG is tested with the application of the different lasers with different linewidths. The optical power input to the resonator keeps consistent while using different light sources. The whole experiment system is established on motionless table.

Fig. 6 Schematic diagram of the resonant fiber optic gyro.

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From the test result shown in Fig. 7, with the application of the fiber lasers FL1 having a relatively small white noise PSD, the detected bias stability of the open loop output fluctuation of the RFOG is approximately 11.58 °/h. The bias stability is about 18.9°/h while using the semiconductor laser LD1 as the light source whose white noise PSD is 33.4 Hz/Hz^1/2. With the application of the semiconductor laser LD2 whose white noise PSD is 250 Hz/Hz^1/2, the bias stability of the open-loop output increases to 204.28 °/h. It is obvious to see that the light sources with a narrow intrinsic linewidth will effectively reduce the intermodulation error in the RFOG system. The larger bias stability with application of FL1 is mainly induced by the drift caused by the nonreciprocal noises in the fiber ring resonator such as double backscattering noise. Further research is necessary to solve this problem.

Fig. 7 Test results of the gyro output with the application of different types of lasers.

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5. Conclusions

This paper for the first time considers the laser frequency noise induced error in the RFOG due to an intermodulation effect. It is worth noting that so long as the modulation and demodulation scheme is implemented, the intermodulation error in the RFOG is inevitable even by choosing a certain modulation waveform or frequency spectrum, because no matter which scheme is implemented, the principle of the resonator information detection through modulation and demodulation is unchanged. The obtainment of the rotation information is based on the detection of the amplitude and phase at different positions of the resonant curve by different modulation frequency spectral component (for instance, between f₀ + f_n and f₀ + f_{n + 1}). The interference component between the modulation frequency spectral components is the demodulation output. If the interference between the modulation frequency spectral components exists, the laser frequency noise can also produces intermodulation with these frequency spectral components and forms intermodulation noise. The generation mechanism of the intermodulation effect is the same as the demodulation output mechanism in the phase modulation detection technique. The utilization of other modulation waveforms, such as square wave or sawtooth wave, also brings about modulation sideband and generates intermodulation noise with the laser frequency noise. The amplitude of the intermodulation noise slightly differs from each other.

Analysis shows that the effect of laser frequency noise on the rotation rate sensitivity should be evaluated by the quantum limited linewidth rather than the spectral linewidth. The laser frequency noise at twice modulation frequency is the dominant source. The rotation rate error caused by the intermodulation effect of a fiber laser with a linewidth of 1 kHz is still one order of magnitude higher than the limited sensitivity of the RFOG system. Light sources with a narrow intrinsic linewidth and a high modulation frequency are preferable to achieve a high rotation-rate sensitivity. The results obtained will be of useful suggestions to the design consideration.

Acknowledgments

The authors would like to acknowledge financial support from the National Natural Science Foundation of China (No. 61377101) for financial support. And thanks to the reviewers from OE for giving the meaningful comments.

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Laser frequency noise induced error in resonant fiber optic gyro due to an intermodulation effect

Abstract

1. Introduction

2. Principle and simulation

2.1 Laser frequency noise

2.2 Principle of the intermodulation effect

2. 3 Analysis and simulation of the intermodulation effect

3. Simulation and analysis

4. Experiments and results

4.1 Simulation of the laser frequency noise by sinusoidal phase modulation

4.2 Application of the lasers with different linewidths to the practical RFOG

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (19)

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