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In this article, we propose an optical heterodyne common-path gyroscope which has common-path configuration and full-dynamic range. Different from traditional non-common-path optical heterodyne technique such as Mach-Zehnder or Michelson interferometers, we use a two-frequency laser light source (TFLS) which can generate two orthogonally polarized light with a beat frequency has a common-path configuration. By use of phase measurement, this optical heterodyne gyroscope not only has the capability to overcome the drawback of the traditional interferometric fiber optic gyro: lack for full-dynamic range, but also eliminate the total polarization rotation caused by SMFs. Moreover, we also demonstrate the potential of miniaturizing this gyroscope as a chip device. Theoretically, if we assume that the wavelength of the laser light is 1550nm, the SMFs are 250m in length, and the radius of the fiber ring is 3.5cm, the bias stability is 0.872 deg/hr.
©2013 Optical Society of America
1. Introduction
The development of gyroscope has been experienced during a long-term evolvement. In 1817, Johann Bohnenberger who was an appointed professor in University of Tübingen proposed the first idea of gyroscope with a mechanical approach. In 1913, Georges Sagnac who was a French physicist proposed the principle of Sagnac effect and setup an optical equipment to find the phase shift caused by a ring interferometer [1]. It is well known that the optical or the electromagnetic approaches for measuring accelerations were more sensitive than mechanical approaches [2]. Over the past thirty years ago, gyroscope still plays an important role in the technology of satellite and national defense. In order to obtain the Sagnac phase critically, scientists and engineers developed many kinds of gyroscope to upgrade the sensitivity and downgrade the bias stability, random walk and scale factor accuracy [3].
In general, the configurations of optical gyroscopes can be divided into four types: ring laser gyroscopes [4,5], interferometric fiber optic gyroscopes (I-FOGs) [6,7], resonance fiber optic gyroscopes (R-FOGs) [8,9], and a photonic crystal based gyroscopes [10,11]. A ring laser gyroscope consists of a ring laser having two counter propagating modes over the same optical path length in order to detect rotatory motion. Although the ring laser gyroscope can serve as a highly sensitive measurement, traditionally, the ring laser gyroscope is too bulky to be used for practical application and too difficult to miniaturize the size of optical setup [12,13]. I-FOG is another kind of ring interferometer which uses multi-turn fiber coils to enhance the Sagnac effect and has been widely used in traditional navigation applications. For guidance applications, a long fiber with length more than 1-2 km usually needs to be used. Unfortunately, such long fiber will result in the increase of the signal drift due to time variant temperature distribution in the sensing coils, which still acts as a barrier in achieving high performance [8,14]. An R-FOG is a resonant approach for this kind of gyroscope. A passive ring cavity is built by use of an optical fiber which is used to instead of the active cavity used in ring laser gyroscope. This configuration is also introduced to enhance the rotation-induced Sagnac effect by a ring resonant cavity. Thanks to the resonant cavity, the length of fiber which used in R-FOG is shorter than I-FOG. As reference indicated, R-FOG usually needs the length of fiber with only 5-10 m [9]. Comparing R-FOG to I-FOG, the amount of usage for fiber is lower and the weight of R-FOG is lighter than I-FOG. However, in order to obtain the best performance, it needs to make more efforts to fabricate the high Q-factor resonator. With the advance of science and technology, in recent years, the photonic crystal has been proposed to apply in gyroscope. B. Z. Steinberg takes use of photonic crystal as a microcavity to enhance the Sagnac phase. He found a set of parameters to obtain the phase shift or frequency difference while the gyroscope suffers from the motion in the directions of counter-clockwise or clockwise. Such photonic-crystal-based gyroscope has the potential to be a compact device [11]. Although R-FOG and photonic crystal based gyroscopes have potential to miniature the size of gyroscope, the manufacturing processing will raise the manufacturing cost. On the other hand, modern fiber optic gyroscope usually uses polarization maintaining fibers (PMFs). In general, PMFs take advantage of two methods to maintain the polarization state of input light wave. One is by use of high expansion glass to induce stress birefringence, and another method is form a special shape of fiber core [15]. Owing to these manufacturing processes will increase the cost, it is more expensive for a gyroscope to use PMFs rather than SMFs.
Optical heterodyne technique had been used to achieve highly sensitive measurement extensively [16–18]. For I-FOG, the optical heterodyne technique is commonly used in interferometry to prevent from the nonlinearity of the cosine response [19–22]. Unfortunately, in order to produce the beat frequency for heterodyne detection, a non-common-path configuration is usually used, such as Mach-Zehnder or Michelson interferometers. These kinds of configurations will be interfered by the change of environment easily and the reciprocity will be destroyed [22]. The non-common-path configuration is the main reason for a heterodyne fiber-optics gyro to increase the bias stability. In recent years, a series of frequency-modulation continuous-wave (FMCW) gyroscopes had been proposed by Zhang [23–25]. FMCW gyroscope use a frequency modulated laser light source to generate a sinusoid wave and use phase sensitive detection technique to obtain the Sagnac phase [26]. However, the difference between this heterodyne gyroscope and FMCW gyroscope is the detection method. This common-path heterodyne gyroscope use amplitude sensitive method to obtain the Sagnac phase. For phase sensitive detection method, it must use phase-demodulation device such as lock-in amplifier (LIA). On the other hand, we can choose LIA or digital voltage multi-meter to acquire the Sagnac signal.
In order to improve the drawback of traditional I-FOG, we develop a heterodyne phase-shifting fiber-optics gyroscope with full-dynamic range and common-path configuration by use of SMFs. In our design of this heterodyne gyroscope, the total polarization rotation caused by SMFs will be eliminated and the use of SMFs will reduce the manufacturing cost. Moreover, this optical setup can also be miniaturized the size and be suitable for actual application.
2. Working principle
2.1 Two-frequency laser light source
In the beginning of the working principle, we will introduce the light source which is used in the proposed setup is a TFLS. The optical setup is illustrated in Fig. 1 [27].
In this figure, P(45þ) represents a polarizer and its transmission axis at 45° respect to the y-axis; EOM(ωt) represents an electro-optic modulator which is modulated by a function generator and a high-voltage driver with a beat frequency ω [28]; OSC is an oscilloscope. As Fig. 1 illustrated, we assume the electric field of incident laser light beam is
Then, the output electric field of TFLS, Ein, can be written in terms of Jones matrix asTheoretically, Eq. (2) shows the TFLS has two orthogonal polarization states with a beat frequency ω. We use this output heterodyne laser light beam to be incident to the SMFs. The experimental result is shown in Fig. 2
Fig. 2 The experimental result of TFLS. The sawtooth wave is generated by a function generator and amplified by a high voltage amplifier. The modulated signal is a Sinusoid wave.
2.2 Heterodyne gyroscope
The optical setup of the heterodyne balanced-detection gyroscope with common-path configuration is shown in Fig. 3.
Fig. 3 The optical setup of the common-path heterodyne gyroscope is illustrated. Where PBS is polarizing beam splitter; BS is beam splitter; EOM is electro-optic modulator; BPF is band-pass filter; HWP is half-wave plate; PR is polarization rotator; DM is amplitude demodulator; PC is personal computer; P is polarizer; D is photodetector.
As Fig. 3 illustrated, the two p- and s- waves of TFLS, which are incident into a multi-turn SMF coils at different ends with different polarization states are divided by a PBS. As a result, the two waves experience a phase shift caused by Sagnac effect respectively. In this study, we aim to calculate the Sagnac phase in terms of Jones calculus [27, 29, 30]. The Jones matrix of this gyroscope G can be expressed as
Whereandare the Jones matrices of the transmission and reflection of a polarizing beam splitter;andare the Jones matrices of the SMFs which the TFLS is incident on different ends. Whereis the average phase shift along the fiber, is the change in ellipticity of the input state of polarization, θ is the total polarization rotation, is the reciprocal phase shift, is the rotation induced phase shift in clockwise direction, is the rotation induced phase shift in counter-clockwise direction.Firstly, we consider the output electric field of p-wave. The Jones vector of the output p-wave can be written as
We use the same method to derive the output s-wave.The p- and s- waves which output from the SMF coils are combined by the same PBS. Soon after, the output p- and s- waves will be divided by a beam splitter (BS). According to Eqs. (8) and (9), in transmission arm, the p- and s-wave will be phase-shifted by EOM 1. Then, we can rewrite these two equations asandIn the arm of reflection, also, the p- and s-wave will be phase-shifted by EOM 2. We assume these two modulated waves in Eqs. (8) and (9) are rewritten asandAs Fig. 3 shows, the p- and s- waves in different arms will be rotated by HWPs (22.5̊) or PR which will make the input p- and s- waves to rotate an angle at 45° with respect to y-axis, and each rotated waves will project to the polarizer P (90̊) with the transmission axis which is parallel to x-axis. In transmission arm:In the same method, the p- and s- waves in the reflection arm can be written asWe use photodetectors to receive the signals of transmission arm and reflection arm respectively. According to Eq. (14), the signal of transmission arm iswhere the Sagnac phase is equal to . The signal of Eq. (16) will be demodulated and filtered by a demodulator and band-pass filter respectively. Then, we will obtainIn the same method, the intensity in reflection arm will be express asAlso, the signal of Eq. (18) will be demodulated and filtered by a demodulator and band-pass filter respectively. Then, we will obtainIf and , we will obtain the intensity from two detectors. According to Eqs. (17) and (19), we haveandSubtracting Eq. (20) from Eq. (21), we haveThen, we add Eq. (20) and Eq. (21). And we obtainThen, dividing Eq. (22) by Eq. (23), we haveIn the same method, if and , we will obtain the intensity from two detectors. We haveandSubtracting Eq. (25) from Eq. (26), we haveAdding Eq. (25) and Eq. (26), we obtainWe divide Eq. (27) by Eq. (28), and then we haveAccording to Eqs. (24) and (29), the optical rotation term will be eliminated. If we divide Eq. (29) by Eq. (24), the Sagnac phase will be obtained asWhere R is the radius of the fiber ring, Δω is the angular velocity, λ is the wavelength of light, and c is the light speed. In Eq. (30), the change in ellipticity of the input state of polarization is a constant, when the length of the SMFs is fixed. We can calculate , before the fiber is wound as a circle. If the phase is known, we can distinguish the signs of the Eqs. (24) and (29). The plus and minus symbols of these two equations can help us to be understand the quadrant of the measured phase which obtained by this heterodyne gyroscope (see Table 1).
3. Analysis of bias stability
The definition of bias stability is the signal output from the gyroscope while it is not experiencing any rotation. In our derivation of this heterodyne gyroscope, as Eq. (30), the relation between the Sagnac phase and the intensities of the two signals is
In Eq. (31), is a constant phase. In order to evaluate the bias stability of this gyroscope, we assume that the intensities of measured long-term drift in steady state (without rotation) are , , and . Then, we can rewrite the Eq. (31) asIf the gyroscope is not experiencing any rotation, the phase of signal is zero. The drift phase can be written asIf the signal is larger than the difference of long-term drifted intensitie, the drift phase can be rewritten asIn practical measurement, are affected by the signals,, and . According to Eq. (34), the bias stability isIf we assume the ratio is 10−6, the wavelength of the laser light is 1550 nm, the SMF is 250 m in length and the radius of the fiber ring is 3.5 cm. It is predictable that the bias stability of this gyroscope is 0.872 deg/hr in this grade of this gyroscope is tactical grade [31].4. Conclusion and discussion
In this theoretical study, the heterodyne fiber-optical gyroscope can provide a common-path configuration and phase shifting measurement with full-dynamic range. The common-path configuration has some features to improve the defects of traditional heterodyne I-FOG [21]. In our calculation, the influence of polarization rotation from SMFs and common noise will be eliminated. However, the length of the SMFs must be designed carefully in order to prevent the rotational angle of SMFs from being equal to 90̊. If the total optical rotation angle of the SMFs is equal to 90̊, the laser light pass through the SMFs will be totally incident on the direction of the laser source. It will degrade the signal-to-noise ratio of this optical heterodyne gyroscope. Moreover, the change in ellipticity of the input polarized state must be obtained in order to calibrate the phase. In addition, the beat length of the SMFs is a key parameter. Generally, the beat length of SMFs is several centimeter. It implies that the accuracy of fiber length must be not more than centimeter.
For practical application, the design of this heterodyne balanced-detection fiber-optical gyroscope with common-path configuration can be miniature as a chip. On the other hand, by use of SMFs, the manufacturing cost will be reduced. As Fig. 4 illustrated, we can combine the technology of integrated-optics polarization rotator [32], polarizers [33] and polarization splitter [34] in Z-propagating Y-cut LiNbO3 to fabricate a chip for this two-frequency heterodyne fiber-optical gyroscope. However, because of SMFs have short beat length, the package process control and temperature control of SMFs must be controlled to make to be a constant. It is predictable that we also are able to combine our basic concept with the idea of high-Q factor passive resonant cavity as a heterodyne R-FOG [35]. This kind of heterodyne R-FOG will have a potential to be a light weight, low cost, and high performance navigation tools.
Fig. 4 The arrangement of the integrated-optic devices on a chip (dash line) for heterodyne fiber-optical gyroscope.
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