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Zero-deviation effect in a resonator optic gyro caused by nonideal digital ramp phase modulation
Optics and Lasers in Engineering 48 (2010) 933–939
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Review
Zero-deviation effect in a resonator optic gyro caused by nonideal digital ramp phase modulation Huilian Ma n, Guhong Zhang, Mucheng Li, Zhonghe Jin Department of Information Science and Electronic Engineering, Yuquan Campus, Zhejiang University, 38 Zheda Road, Hangzhou City 310027, PR China
a r t i c l e in fo
abstract
Article history: Received 24 March 2010 Received in revised form 4 May 2010 Accepted 31 May 2010 Available online 16 June 2010
Resonator fiber optic gyro (RFOG) is a high accuracy inertial rotation sensor based on the Sagnac effect. The digital ramp phase modulation technique is usually adopted instead of analog modulation to improve accuracy and stability. Failure to keep the stair-period of the digital ramp waveform equal to the optical transmission time in the fiber ring resonator produces errors in the RFOG. The influence of the nonideal stair period on gyro performance is firstly fully developed in this paper. The physical mechanism is an uncertain phase relationship between the transmitted light and the circulated light components in the resonator occurring in this nonideal stair-period. The effect on output signals of the resonator is fully analyzed. Through simulation, it is found that the zero position of the demodulation curve in the symmetrically digital triangle phase modulation technique is not affected by the nonideal stair-period problem. However, in the two-frequency combined digital serrodyne phase modulation technique, much effort is needed to overcome the zero-deviation, which causes errors in the RFOG. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Resonator optic gyro Digital ramp phase modulation Zero-deviation
Contents 1. 2. 3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation and analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Influence of the output signal of the OFRR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Influence of the resonance curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Zero-deviation of the demodulation curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction The first implementation of the Sagnac effect by the use of a passive resonator was proposed and demonstrated early in 1977 by Ezekiel and Balsamo [1]. Compared with the very successful He–Ne ring laser gyro (RLG), resonator fiber optic gyro (RFOG) based on the passive resonator does not have the gain competition problem. Compared with the interferometric fiber optic gyro (IFOG) [2,3], RFOG has the potential for realizing IFOG-like performance with a coil length up to 100 shorter than that of IFOG for a given performance class [4]. However, it is clear that much effort is needed to make RFOG viable [5].
n
Corresponding author. Tel./fax: + 86 571 8795 2587. E-mail address: [email protected] (H. Ma).
0143-8166/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2010.05.012
933 934 935 935 936 938 939 939 939
In the RFOG, the rotation rate readout is given as the resonance frequency difference between the clockwise (CW) and counterclockwise (CCW) lightwaves propagating in the optical fiber ring resonator (OFRR) through the Sagnac effect [1]. The frequency difference is measured by external means. Various signal detection schemes are proposed and demonstrated [6–9]. In the frequency modulation technique [7], the modulating signal and the feedback signal are both applied to the laser source, which causes interference between them. The phase modulation spectroscopy technique first suggested for the RFOG by Sanders et al. [7] does not have the interference problem. It can be applied with an electro-optic modulator, which is helpful to reduce the system size, and has large flexibility compared with that applied with a piezoelectric transducer (PZT) modulator [9]. Two different waveforms including the sinusoidal and serrodyne (sawtooth with instantaneous fly-back) waveforms are usually adopted in the phase modulation spectroscopy technique.
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H. Ma et al. / Optics and Lasers in Engineering 48 (2010) 933–939
For a closed-loop operation RFOG system, an optical frequency shifter is required to track the resonant state of the OFRR. Frequency shifting occurs when the phase of the light is modulated by a serrodyne waveform with the amplitude of 2p. The optical frequency shift equals the frequency of the serrodyne waveform [10]. Compared with the sinusoidal waveform, the serrodyne waveform possesses both of the functions of modulation and frequency shift. Strandjord and Sanders [11]. have analyzed the error in rotation rate sensitivity caused by the serrodyne amplitude deviating from 2p. Hotate and Harumoto [12] have digitalized the serrodyne phase modulation technique, which makes it easy to track the resonant frequency with high accuracy. It is shown that when the stair period of the digital ramp phase waveform is the same as the optical transmission time in the OFRR, the digital serrodyne waveform is accurately equivalent to the ideal analog serrodyne waveform. In order to further analyze and better design the RFOG system, deep analysis of the influence of nonideal stair period of the digital ramp phase waveform on gyro performance is fully first developed in this paper. The output optical field of the OFRR is the superposition among the input light transmitted through the resonator coupler and the multiple circulated light components in the OFRR. Through analyzing it is found that the source of the error is the uncertain phase difference between the transmitted light and circulated light components in the OFRR. On this basis, the effect on the output signal of the OFRR with a fixed laser frequency, the resonance curves of the OFRR and the demodulation curves are fully analyzed. The zero-deviation of the demodulation curve causes errors in the rotation rate readout directly. Through simulation, it is found that the zero position of the demodulation curve in the symmetrically digital triangle phase modulation with frequency shifts + F and F [13,14] is not affected by this nonideal stair period. However, in the twofrequency combined digital serrodyne phase modulation waveform, much effort is needed to overcome the zero-deviation problem, which causes errors in the RFOG.
2. Theory Fig. 1 illustrates the system configuration of the RFOG based on the digital ramp phase modulation technique. All the fibers in the system are polarization maintaining. The intensity coupling coefficient for the couplers C1, C2, and C3 are all designed as 0.5. The intensity coupling coefficient for the coupler C4 is 0.1. The center wavelength and linewidth of the fiber laser are 1550 and 60 kHz, respectively. Light is divided into two beams by the coupler C1 and each beam is digitally ramp phase modulated by the LiNbO3 phase modulators PM1 and PM2 before being launched into the OFRR. The CW and CCW beams in the OFRR
are detected by the InGaAs PIN photodetectors PD1 and PD2, respectively. The digital demodulation circuit DMC1 demodulates the CW signal, which is used as an error signal in the feedback circuit (FBC) to lock the laser frequency to the resonant frequency of the CW lightwave in the OFRR. When the demodulation signal of DMC1 is positive, a negative feedback is produced by the FBC. When the demodulation signal of DMC1 is negative, a positive feedback is produced by the FBC. The demodulated signal of CCW lightwave is used as the open-loop output of the rotation rate. Any zero-deviation of the demodulation curve causes the rotation rate readout directly. The output field of the fiber laser is written as Elaser ðtÞ ¼ E0 exp ið2pf0 t þ j0 Þ ð1Þ where E0 is the amplitude of the laser, f0 the center frequency of the laser, and j0 the initial phase. When the phase of the CW beam is digitally ramp modulated before being launched into the OFRR, the field at the entrance of the OFRR is written as Ein ðtÞ ¼
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1aC1 Þð1aC3 Þð1aPM2 ÞE0 exp i 2pf0 t þ jðtÞ þ j0 2 ð2Þ
where aC1 and aC3 are the fractional intensity loss of the couplers C1 and C3, respectively, aPM2 is the insertion loss of the phase modulator PM2, and j(t) the phase induced by the digital ramp phase waveform. The light transmitted through the directional coupler C4 directly, Ethr(t), is written as ð3Þ Ethr ðt Þ ¼ kthr E0 exp i 2pf0 t þ jðtÞ þ j0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kthr ¼ ð1aC1 Þð1aC3 Þð1aPM2 Þð1kC Þð1aC4 Þ=2; kC and aC4 are the intensity coupling coefficient and fractional intensity loss of the directional coupler C4, respectively. The output light transmitted through the directional coupler C4 after circulating in the OFRR for n (n is an integer) turns is written as [15,16] Encro ðt Þ ¼ kncro E0 exp i 2pf0 ðtntÞ þ jðtntÞ þ j0 þ p ð4Þ h i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 where kncro ¼ kthr kC ð1aL Þð1aC4 Þ ð1kC Þð1aC4 Þð1aL Þ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1kC Þ, aL is the fractional intensity loss of the OFRR, t ¼nrL/c the transit time in the OFRR, L the fiber length of the OFRR, nr the refractive index of the fiber, and c the light velocity in vacuum. The optical field at the exit port of the OFRR is given by the superposition among the input light transmitted through the coupler and the multiple circulated light components in the OFRR written as
Eout ðtÞ ¼ Ethr ðtÞ þ
1 X
Encro ðtÞ
ð5Þ
n¼1
Fig. 1. System configuration of RFOG based on the digital ramp phase modulation technique. FL: fiber laser; C1, C2, C3, and C4: couplers; OFRR: optical fiber ring resonators; PD1 and PD2: photodetectors; DMC1 and DMC2: demodulation circuits; PM1 and PM2: phase modulators; FBC: feedback circuit.
H. Ma et al. / Optics and Lasers in Engineering 48 (2010) 933–939
waveform:
The output signal at the photodetector PD1 is written as D 2 E 1 VPD ¼ aC2 NPD Eout ðtÞ 2
935
ð6Þ
where aC2 is the fractional intensity loss of the coupler C2 and, NPD the photoelectric transformation coefficient of the photodetector PD1. The demodulation signal is obtained by the crosscorrelation of VPD and the synchronizing square signal of the digital serrodyne waveform [12,13].
F¼
1 Dy 2p tu
ð7Þ
Introduce a proportionality coefficient p to express the relationship between t0 and the transmission time t in the OFRR:
tu ¼ pt
ð8Þ
Correspondingly, the phase shift for one stair is denoted as Dyp.
Dyp ¼ p Dy1 3. Simulation and analysis The general simulation parameters are as follows: the fiber length L of the OFRR is 10 m, the refractive index nr of the fiber is 1.45, the intensity coupling coefficient kC of the coupler C4 is 0.1, the excess losses aC1, aC2, aC3, and aC4 for the couplers C1, C2, C3, and C4, respectively, are all designed as 0.4 dB, the insert loss of the phase modulators PM1 and PM2 are 50%, the total propagation loss aL of the OFRR is 0.05 dB, the output power of the fiber laser is 1 mW, the gain of the demodulation circuit is 1, and the photoelectric transformation coefficient NPD is 6.25 V/mW.
ð9Þ
where Dy1 is the phase shift for one stair when the stair-period of the digital ramp phase waveform is t, i.e. p in Eq. (8) equals 1. Set the instant phase induced by the digital ramp phase waveform for Ethr(t) as j(t), if p ¼1, then j(t) can be written as
jðtÞ ¼ m Dy1
ð10Þ
where m is an integer. The corresponding phase induced by the digital ramp phase waveform for Encro ðtÞ can be written as
jðtntÞ ¼ ðmnÞDy1
ð11Þ
3.1. Influence of the output signal of the OFRR
If pa1, that is to say the stair-period is not equivalent to the transmission time in the OFRR then the phase induced by the digital ramp phase for Ethr(t) and Encro ðtÞ can be written as m þ1 m þ 1n jðtÞ ¼ Dyp , jðtntÞ ¼ Dyp ð12Þ p p int int
The digital ramp voltage applied with the phase modulator achieves the optical frequency shift. Phase of the lightwave is modulated by f(t) as shown in Fig. 2 before being launched into the OFRR. A general two-frequency combined digital serrodyne phase modulation waveform alternating between the two different slopes of F1 and F2 with the repetition rate of q is shown in Fig. 2(a). A special case for the symmetrically digital triangle phase waveform alternating between the positive and negative slopes with the repetition rate of F/2 is shown in Fig. 2(b); t0 is the stair period of the digital ramp phase waveform and Dy the phase shift for one stair. F is defined as the slope of the
where [ ]int is the largest integer smaller than the value in the bracket. When p is an integer greater than 1, the circulating fields achieve a phase shift of Dyp only after circulating in the OFRR for p turns, and p successive components share the same phase shift. For every cross field Encro ðtÞ circulated in the OFRR n turn, it has p possible phase shifts. Table 1 shows the phase shift introduced by the digital ramp waveform for Ethr(t) and Encro ðtÞ in the case of p¼1 and 2, respectively. As seen from Table 1, there are two possible phase shifts for the optical field Encro ðtÞ with n being an odd number. Fig. 3 shows the output signal of PD1 when the laser frequency f0 equals the resonant frequency fc in the case of p¼1, 2, 10, 1.1, 1.8, and 2.1. The frequency shift of the digital serrodyne phase
(t)
2π repetition frequency q
′ 1/F1
1/F2 t
(t)
2π
′ 1/F
′ 1/F
1/F
1/F
t Fig. 2. Digital ramp phase waveforms. (a) A general two-frequency combined digital serrodyne phase modulation waveform alternating between the two different slopes of F1 and F2 with the repetition rate of q and (b) a special case for the symmetrically digital triangle phase waveform alternating between the positive and negative slope with the repetition rate of F/2.
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H. Ma et al. / Optics and Lasers in Engineering 48 (2010) 933–939
Table 1 Phase shift introduced by the digital ramp phase waveform for Ethr(t) and Encro ðtÞ for the case of p ¼ 1 and 2, respectively. Optical field
j(t) phase shift induced by the digital ramp phase waveform and Dyp phase shift for one step
Phase shift
j(t)(p ¼ 1) ot+ j0 + j(t) ot+ j0 ot + j (t t) o + j0t 2ot + j (t 2t) ot+ j0 3ot + j (t 3t) ot+ j0 4ot + j (t 4t)
Ethr E1cro E2cro E3cro E4cro ^ Encro
j(t)(p ¼ 2) m Dy2
m Dy1 (m 1)Dy1 (m 2)Dy1 (m 3)Dy1 (m 4)Dy1 ^ (m n)Dy1
^
ot+ j0 not + j (t nt)
m Dy2 Dy2 m Dy2 Dy2 m Dy2 2Dy2 m Dy2 2Dy2 ^ nþ1 m Dy2 Dy2 2 int
m Dy2 m Dy2 Dy2 m Dy2 Dy2 m Dy2 2Dy2 h ni m Dy2 Dy2 2 int
m is an integer.
0.09
0.09
0.20
0.07 0.06
Intensity at PD1
Intensity at PD1
Intensity at PD1
0.08
0.08 0.07 0.06 0.05
0.05 0
1
2 3 t (μsec)
4
1
2 3 t (μsec)
4
0
1
2 3 t (μsec)
0.09
0.06 0.05
Intensity at PD1
Intensity at PD1
Intensity at PD1
0.07
0.08 0.07 0.06
5
4
5
p = 2.1
0.08 0.07 0.06 0.05
0.05 4
0.05
p = 1.8
0.08
2 3 t (μsec)
0.10
5
0.09 p = 1.1
1
0.15
0.00 0
5
0.09
0
p = 10
p=2
p=1
0
1
2 3 t (μsec)
4
5
0
1
2 3 t (μsec)
4
5
Fig. 3. Output signal of the FORR in the case of p ¼ 1, 2, 10, 1.1, 1.8, and 2.1.
waveform is 100 kHz. As seen from Fig. 3, the output signal of PD1 has a steady value with p¼1. In the case of p¼2 and 10, respectively, the output signal of PD1 is interchanging between two and ten different values. As p increases, the possible phase shifts of the cross-fields increase, so the outputs are interchanging between p different values. When the reciprocal of p is an integer greater than 1, the circulating fields achieve 1/p multiples of Dyp after circulating the FORR for one turn. The output of PD1 is kept at a steady value, the same as with the case of p¼ 1. When p or 1/p is not an integer, the phase shift of the circulating field is more complex. The output field fluctuates circulating around the ideal value with the period of t. The output values are sampled with the period of t; after averaging, there is still some deviation between the mean value and the ideal value. It can be further found that the deviation is dependent on p. When p is deviated further off one, the deviation of the output is bigger.
frequency of the lightwave launching into the OFRR, the resonance curve of OFRR is observed at PD. Fig. 4 shows the resonance curves with the frequency range of one free spectral range (FSR) in the case of p¼ 1, 2.5, and 10. Here FSR ¼c/nrL. As seen from Fig. 4, when p is an integer greater than 1, there are p periods duration of resonance curves in the frequency range of one FSR. It seems that the frequency distance between two neighboring resonance dips is compressed as 1/p of FSR. According to Eq. (7), when the phase shift of one stair of digital ramp phase waveform satisfies
Dyp
tu
¼
2p
t
the frequency shift induced by the digital ramp phase waveform is one FSR. Combining Eq. (8) and (13), Dyp satisfies
Dyp ¼ p2p 3.2. Influence of the resonance curve Setting the frequency of fiber laser equal to the resonant frequency of the OFRR and changing the slope of the digital ramp phase waveform linearly with time is used to sweep the
ð13Þ
ð14Þ
Sweeping the frequency of the digital ramp phase waveform for one FSR, the phase shift Dyp of one stair should equal 2pp. In fact, one resonance dip comes out for every phase shift of 2p. Pseudo-resonance dips come out when p is bigger than 1 as shown in Fig. 4. In order to avoid the influence of the
H. Ma et al. / Optics and Lasers in Engineering 48 (2010) 933–939
p=1 p = 2.5 p = 10
1
937
0.6 p = 20
0.9
Intensity at PD1
0.7 0.6 Iout
p = 10 p=1
0.5
0.8
0.5 0.4
0.4 0.3 0.2
0.3
0.1
0.2 0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 frequency of digital serrodyne waveform *FSR
0.0 -300
-200
-100
0.5
0 100 frequency (kHz)
300
0.6
Fig. 4. Resonance curves of the OFRR in the case of p ¼ 1, 2.5, and 10.
p = 20 p = 10 p=1
0.5
Intensity at PD1
p=1 p = 0.8 p = 0.5
1
200
0.9 0.8 0.7
0.4 0.3 0.2
Iout
0.6 0.1
0.5 0.4
0.0 -400
0.3
-300
-200
0 -100 frequency (kHz)
100
200
0.2 0.1
0.6 p = 20 p = 10 p=1
0 1
1.5
frequency of digital serrodyne waveform
-1.5
-1
-0.5
0
0.5
*FSR
2
0.5
Fig. 5. Resonance curves for the frequency range of four FSR in the case of p ¼1, 0.8, and 0.5.
0.4
pseudo-resonance dips, the slope of the digital ramp phase waveform should be lower than 2p FSR/p. Fig. 5 shows the resonance curves for the frequency range of four FSRs in the case of p¼1, 0.8, and 0.5. When p is less than 1, the frequency distance between two neighboring resonance dips is elongated 1/p times of FSR. Only one cycle of the resonance curve appears in 1/p times of FSR range. Next we will discuss the influence of the position of the resonance dip. Fig. 6 shows the resonance curves around the resonance dip while sweeping the frequency shift of the digital ramp phase waveform in the case of p ¼1, 10, and 20, respectively. Fig. 6(a) shows the resonance curve for f0 ¼fc. Fig. 6(b) shows the results for f0 ¼fc +100 kHz and Fig. 6(c) shows the results for f0 ¼fc 100 kHz. Solid line for p ¼20, dash line is for p ¼10, and dot line for p¼ 1. As seen from Fig. 6, for f0 ¼ fc, the resonance curves are apart for different values of p but they have the same resonance dips. For f0 ¼fc + 100 kHz, the resonance curves are upshifted and the resonance dip has moved rightwards as p
Intensity at PD1
-2
0.3 0.2 0.1 0.0 -200
-100
0
200 100 frequency (kHz)
300
400
Fig. 6. Resonance curves around the resonance dip in the case of p ¼1, 10, and 20. (a) f0 ¼fc; (b) f0 ¼ fc + 100 kHz; and (c) f0 ¼fc 100 kHz.
increases. For f0 ¼ fc 100 kHz, the resonance curves are upshifted too; however the resonance dip has moved leftwards as p increases. The movement of the resonance dip affects the lockin position of the center frequency of the laser. This means that the laser frequency is not locked to the practical resonant
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H. Ma et al. / Optics and Lasers in Engineering 48 (2010) 933–939
frequency, which affects the rotation rate readout directly. Through analyzing the simulation results in Fig. 6, it can be found that there is no deviation of the resonance dip when the laser frequency is the same as the resonant frequency. That is to say, if the lock-in state corresponds to f0 ¼fc, the laser frequency is locked to the practical resonant point in spite of any values of p. The error in the resonant frequency (Dfr) is dependent on p and the difference between the laser frequency and the resonant frequency (Df¼f0 fc). Fig. 7 shows the relationship between the error in the resonant frequency and the resonant frequency deviation Df. Two cases with p of 0.8 and 2 are calculated. Dividing Dfr by Df introduces a unit error in the resonant frequency Dfr0. Then Dfr can be written as
Dfr ¼ Dfr0 Df
ð15Þ
Eq. (15) indicates that when Df is 0, Dfr is kept at zero in spite of any values of Dfr0. Fig. 8 shows the relationship between the unit resonant frequency deviation Dfr0 and p. As seen from Fig. 8, there is no error for the ideal stair period (p ¼1). Errors appear for both decimal and integer numbers of p expect for the ideal stair period (p ¼1) as shown in Fig. 8. When p is closer to 1, the resonant frequency deviation is smaller.
3.3. Zero-deviation of the demodulation curve Previous analysis is done only for a single digital ramp phase waveform applied with the phase modulator in the RFOG. A combined waveform with two different slopes is used to detect the gyro signal in the practical RFOG. For a two-frequency combined serrodyne modulation waveform [12], the twofrequency shifts are F1 and F2. For the symmetric triangle waveform, the two-frequency shifts are F and F. Three cases of p¼ 1, p ¼10, and p ¼20 are calculated. Fig. 9 shows the demodulation curves for the digital triangle phase waveform with the frequency shifts 80 and 80 kHz applied with the phase modulator. The lock-in state is f0 + (80 80) kHz/2¼fc, according to Eq. (15), Dfr is always zero in the case of Df¼ 0 in spite of any values of p. So the laser frequency is locked to the resonant frequency and the demodulation output is zero. There is no zero-deviation of the demodulation curve regardless of any values of p. The same result is shown in Fig. 9. Though there is some difference in the amplitude of the resonance curves as shown in Fig. 6, amplitudes of the three demodulation curves are very similar. The good symmetry in the resonance curves around the resonance dip counteracts the difference in the amplitude of the
0.40
p=1 p = 10 p = 20
0.35
p=2
0.2 0.15
0.30 demodulation value
error in the resonant frequency /Hz
0.25
0.25 0.20 0.15 0.10
0.1 0.05 0 -0.05 -0.1 -0.15
p = 0.8
-0.2
0.05
-0.25
0.00 40
60
80
100 f0-fc/Hz
140
120
-0.05 -0.04 -0.03 -0.02 -0.01
0
frequency deviation
Fig. 7. Relationship between the error in the resonant frequency (Dfr) and the resonance frequency deviation (f0 fc) in the case of p ¼ 0.8 and 2, respectively.
*FSR
Fig. 9. Demodulation curve for the symmetrically digital triangle phase waveform with frequency shifts 80 and 80 kHz.
0.1
0.0030
p=1 p = 10 p = 20
0.08
0.0025
0.06 demodulation value
error in the unit resonance frequency
0.01 0.02 0.03 0.04 0.05
0.0020 0.0015 0.0010 0.0005
0.04
6kHz
0.02 22kHz
0 -0.02 -0.04 -0.06 -0.08
0.0000 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
p Fig. 8. Relationship between the error in the unit resonance frequency (Dfr0) with p.
-0.1 -0.05 -0.04 -0.03 -0.02 -0.01
0
frequency deviation
0.01 0.02 0.03 0.04 0.05 *FSR
Fig. 10. Demodulation curve for the two-frequency combined digital serrodyne phase waveform with frequency shifts 50 and 100 kHz.
H. Ma et al. / Optics and Lasers in Engineering 48 (2010) 933–939
demodulation curves. Fig. 10 shows the demodulation curve for the two-frequency combined digital serrodyne phase modulation waveform with the frequency shifts 50 and 100 kHz applied with the phase modulator. The locked condition is f0 + (50+ 100) k/2 ¼fc, i.e Df ¼ 75 kHz, so there is a zero-deviation of the demodulation curve in the case of p a1. The zero-deviation is about 6 and 22 kHz in the case of p¼10 and p¼20, respectively. For an OFRR with a diameter of 10 cm, the gyro scale factor is about 44 kHz/ (rad/s). This big zero-deviation needs to be avoided.
4. Conclusion Errors in RFOG based on the digital ramp phase modulation technique are analyzed. The physical mechanism for the error is the uncertain phase difference between the transmitted light and circulated light components in the OFRR. The zero-deviation of the demodulation curves in the symmetrically digital triangle and the two-frequency combined serrodyne phase modulation waveforms are compared. It is found that the zero-position of the demodulation curve in the symmetrically digital triangle phase modulation technique is not affected by the nonideal stair-period problem. In other words, the symmetrically digital triangle wave phase modulation technique is more advantageous in reducing the nonideal stair-period problem. The theoretical results are helpful to optimize the parameters in the digitalized RFOG system.
Acknowledgement The authors thank the Specialized Research Fund for the Doctoral Program of High Education, PR China (20060335064) for financial support.
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References [1] Ezekiel S, Balsamo SR. Passive ring resonator laser gyroscope. Appl. Phys. Lett. 1977;30:478–80. [2] George D, Pavlath A. Fiber optic gyros: the vision realized. In: Proceeddings of the 18th international optical fiber sensors conference technical digest, Optical Society of America, Washington, DC, 2006 [MA3.pdf]. [3] Ohno A, Kurokawa A, Kumagai T, et al. Applications and technical progress of fiber optic gyros in Japan. In: Proceedings of the 18th international optical fiber sensors conference technical digest, Optical Society of America, Washington, DC 2006 [MA4.pdf]. [4] Ezekiel S. Optical gyroscope options: principles and challenges. In: Proceedings of the 18th international optical fiber sensors conference technical digest, Optical Society of America, Washington, DC 2006 [MC1.pdf]. [5] Sanders GA, Strandjord LK, Qiu T. Hollow core fiber optic ring resonator for rotation sensing. In: Proceedings of the 18th international optical fiber sensors conference technical digest, Optical Society of America, Washington, DC 2006 [ME6.pdf]. [6] Imai T, Nishide KI, Ochi H, Ohtsu M. The passive ring resonator fiber optic gyro using modulatable highly coherent laser diode module. Proc Soc PhotoOpt Instrum Eng 1991;158: 153–62. [7] Sanders GA, Prentiss MG, Ezekiel S. Passive ring rotation method for sensitive inertial rotation measurements in geophysics and relativity. Opt Lett 1981;6:569–71. [8] Carroll R, Coccoli CD, Cardarelli D, Coate GT. Passive resonator fiber optic gyro and comparison to the interferometer fiber gyro. Proc Soc Photo-Opt Instrum Eng 1986; 719: 169–76. [9] Yi Y, Shi K, Lu W, Jian S. Phase modulation spectroscopy for resonator fiber gyroscope by using all-fiber PZT phase modulator. Appl Opt 1995;34:7383–6. [10] Sanders GA, Rouse GF, Strandjord LK, et al. Resonator fiber-optic gyro using LiNbO3 integrated optics at 1.5 mm wavelength. Proc Soc Photo-Opt Instrum Eng, 1988; 985: 202–10. [11] Strandjord LK, Sanders GA. Effects of imperfect serrodyne phase modulation in resonator fiber optic gyroscopes. Proc Soc Photo-Opt Instrum Eng 1994;2292:272–82. [12] Hotate K, Harumoto M. Resonator fiber optic gyro using digital serrodyne modulation. J of Lightwave Technol 1997;15:466–73. [13] Ying DQ, Ma HL, Jin ZH. Resonator fiber optic gyro using the triangle wave phase modulation technique. Opt Commun 2008;281:580–6. [14] Jin ZH, Yang ZH, Ma HL, Ying DQ. Open-loop experiments in a resonator fiberoptic gyro using digital triangle wave phase modulation. IEEE Photon Technol Lett 2007;19:1685–7. [15] Ioannidis ZK, Radmore PM, Giles IP. Dynamic response of an all-fiber ring resonator. Opt Lett 1988;13:422–4. [16] Ma HL, Jin ZH, Ding C, Wang YL. Influence of spectral linewidth of laser on resonance characteristics in fiber ring resonator. Chin J Lasers 2003;30: 731–4.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Review
Zero-deviation effect in a resonator optic gyro caused by nonideal digital ramp phase modulation Huilian Ma n, Guhong Zhang, Mucheng Li, Zhonghe Jin Department of Information Science and Electronic Engineering, Yuquan Campus, Zhejiang University, 38 Zheda Road, Hangzhou City 310027, PR China
a r t i c l e in fo
abstract
Article history: Received 24 March 2010 Received in revised form 4 May 2010 Accepted 31 May 2010 Available online 16 June 2010
Resonator fiber optic gyro (RFOG) is a high accuracy inertial rotation sensor based on the Sagnac effect. The digital ramp phase modulation technique is usually adopted instead of analog modulation to improve accuracy and stability. Failure to keep the stair-period of the digital ramp waveform equal to the optical transmission time in the fiber ring resonator produces errors in the RFOG. The influence of the nonideal stair period on gyro performance is firstly fully developed in this paper. The physical mechanism is an uncertain phase relationship between the transmitted light and the circulated light components in the resonator occurring in this nonideal stair-period. The effect on output signals of the resonator is fully analyzed. Through simulation, it is found that the zero position of the demodulation curve in the symmetrically digital triangle phase modulation technique is not affected by the nonideal stair-period problem. However, in the two-frequency combined digital serrodyne phase modulation technique, much effort is needed to overcome the zero-deviation, which causes errors in the RFOG. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Resonator optic gyro Digital ramp phase modulation Zero-deviation
Contents 1. 2. 3.
4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation and analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Influence of the output signal of the OFRR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Influence of the resonance curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Zero-deviation of the demodulation curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction The first implementation of the Sagnac effect by the use of a passive resonator was proposed and demonstrated early in 1977 by Ezekiel and Balsamo [1]. Compared with the very successful He–Ne ring laser gyro (RLG), resonator fiber optic gyro (RFOG) based on the passive resonator does not have the gain competition problem. Compared with the interferometric fiber optic gyro (IFOG) [2,3], RFOG has the potential for realizing IFOG-like performance with a coil length up to 100 shorter than that of IFOG for a given performance class [4]. However, it is clear that much effort is needed to make RFOG viable [5].
n
Corresponding author. Tel./fax: + 86 571 8795 2587. E-mail address: [email protected] (H. Ma).
0143-8166/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2010.05.012
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In the RFOG, the rotation rate readout is given as the resonance frequency difference between the clockwise (CW) and counterclockwise (CCW) lightwaves propagating in the optical fiber ring resonator (OFRR) through the Sagnac effect [1]. The frequency difference is measured by external means. Various signal detection schemes are proposed and demonstrated [6–9]. In the frequency modulation technique [7], the modulating signal and the feedback signal are both applied to the laser source, which causes interference between them. The phase modulation spectroscopy technique first suggested for the RFOG by Sanders et al. [7] does not have the interference problem. It can be applied with an electro-optic modulator, which is helpful to reduce the system size, and has large flexibility compared with that applied with a piezoelectric transducer (PZT) modulator [9]. Two different waveforms including the sinusoidal and serrodyne (sawtooth with instantaneous fly-back) waveforms are usually adopted in the phase modulation spectroscopy technique.
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For a closed-loop operation RFOG system, an optical frequency shifter is required to track the resonant state of the OFRR. Frequency shifting occurs when the phase of the light is modulated by a serrodyne waveform with the amplitude of 2p. The optical frequency shift equals the frequency of the serrodyne waveform [10]. Compared with the sinusoidal waveform, the serrodyne waveform possesses both of the functions of modulation and frequency shift. Strandjord and Sanders [11]. have analyzed the error in rotation rate sensitivity caused by the serrodyne amplitude deviating from 2p. Hotate and Harumoto [12] have digitalized the serrodyne phase modulation technique, which makes it easy to track the resonant frequency with high accuracy. It is shown that when the stair period of the digital ramp phase waveform is the same as the optical transmission time in the OFRR, the digital serrodyne waveform is accurately equivalent to the ideal analog serrodyne waveform. In order to further analyze and better design the RFOG system, deep analysis of the influence of nonideal stair period of the digital ramp phase waveform on gyro performance is fully first developed in this paper. The output optical field of the OFRR is the superposition among the input light transmitted through the resonator coupler and the multiple circulated light components in the OFRR. Through analyzing it is found that the source of the error is the uncertain phase difference between the transmitted light and circulated light components in the OFRR. On this basis, the effect on the output signal of the OFRR with a fixed laser frequency, the resonance curves of the OFRR and the demodulation curves are fully analyzed. The zero-deviation of the demodulation curve causes errors in the rotation rate readout directly. Through simulation, it is found that the zero position of the demodulation curve in the symmetrically digital triangle phase modulation with frequency shifts + F and F [13,14] is not affected by this nonideal stair period. However, in the twofrequency combined digital serrodyne phase modulation waveform, much effort is needed to overcome the zero-deviation problem, which causes errors in the RFOG.
2. Theory Fig. 1 illustrates the system configuration of the RFOG based on the digital ramp phase modulation technique. All the fibers in the system are polarization maintaining. The intensity coupling coefficient for the couplers C1, C2, and C3 are all designed as 0.5. The intensity coupling coefficient for the coupler C4 is 0.1. The center wavelength and linewidth of the fiber laser are 1550 and 60 kHz, respectively. Light is divided into two beams by the coupler C1 and each beam is digitally ramp phase modulated by the LiNbO3 phase modulators PM1 and PM2 before being launched into the OFRR. The CW and CCW beams in the OFRR
are detected by the InGaAs PIN photodetectors PD1 and PD2, respectively. The digital demodulation circuit DMC1 demodulates the CW signal, which is used as an error signal in the feedback circuit (FBC) to lock the laser frequency to the resonant frequency of the CW lightwave in the OFRR. When the demodulation signal of DMC1 is positive, a negative feedback is produced by the FBC. When the demodulation signal of DMC1 is negative, a positive feedback is produced by the FBC. The demodulated signal of CCW lightwave is used as the open-loop output of the rotation rate. Any zero-deviation of the demodulation curve causes the rotation rate readout directly. The output field of the fiber laser is written as Elaser ðtÞ ¼ E0 exp ið2pf0 t þ j0 Þ ð1Þ where E0 is the amplitude of the laser, f0 the center frequency of the laser, and j0 the initial phase. When the phase of the CW beam is digitally ramp modulated before being launched into the OFRR, the field at the entrance of the OFRR is written as Ein ðtÞ ¼
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1aC1 Þð1aC3 Þð1aPM2 ÞE0 exp i 2pf0 t þ jðtÞ þ j0 2 ð2Þ
where aC1 and aC3 are the fractional intensity loss of the couplers C1 and C3, respectively, aPM2 is the insertion loss of the phase modulator PM2, and j(t) the phase induced by the digital ramp phase waveform. The light transmitted through the directional coupler C4 directly, Ethr(t), is written as ð3Þ Ethr ðt Þ ¼ kthr E0 exp i 2pf0 t þ jðtÞ þ j0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kthr ¼ ð1aC1 Þð1aC3 Þð1aPM2 Þð1kC Þð1aC4 Þ=2; kC and aC4 are the intensity coupling coefficient and fractional intensity loss of the directional coupler C4, respectively. The output light transmitted through the directional coupler C4 after circulating in the OFRR for n (n is an integer) turns is written as [15,16] Encro ðt Þ ¼ kncro E0 exp i 2pf0 ðtntÞ þ jðtntÞ þ j0 þ p ð4Þ h i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 where kncro ¼ kthr kC ð1aL Þð1aC4 Þ ð1kC Þð1aC4 Þð1aL Þ = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1kC Þ, aL is the fractional intensity loss of the OFRR, t ¼nrL/c the transit time in the OFRR, L the fiber length of the OFRR, nr the refractive index of the fiber, and c the light velocity in vacuum. The optical field at the exit port of the OFRR is given by the superposition among the input light transmitted through the coupler and the multiple circulated light components in the OFRR written as
Eout ðtÞ ¼ Ethr ðtÞ þ
1 X
Encro ðtÞ
ð5Þ
n¼1
Fig. 1. System configuration of RFOG based on the digital ramp phase modulation technique. FL: fiber laser; C1, C2, C3, and C4: couplers; OFRR: optical fiber ring resonators; PD1 and PD2: photodetectors; DMC1 and DMC2: demodulation circuits; PM1 and PM2: phase modulators; FBC: feedback circuit.
H. Ma et al. / Optics and Lasers in Engineering 48 (2010) 933–939
waveform:
The output signal at the photodetector PD1 is written as D 2 E 1 VPD ¼ aC2 NPD Eout ðtÞ 2
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ð6Þ
where aC2 is the fractional intensity loss of the coupler C2 and, NPD the photoelectric transformation coefficient of the photodetector PD1. The demodulation signal is obtained by the crosscorrelation of VPD and the synchronizing square signal of the digital serrodyne waveform [12,13].
F¼
1 Dy 2p tu
ð7Þ
Introduce a proportionality coefficient p to express the relationship between t0 and the transmission time t in the OFRR:
tu ¼ pt
ð8Þ
Correspondingly, the phase shift for one stair is denoted as Dyp.
Dyp ¼ p Dy1 3. Simulation and analysis The general simulation parameters are as follows: the fiber length L of the OFRR is 10 m, the refractive index nr of the fiber is 1.45, the intensity coupling coefficient kC of the coupler C4 is 0.1, the excess losses aC1, aC2, aC3, and aC4 for the couplers C1, C2, C3, and C4, respectively, are all designed as 0.4 dB, the insert loss of the phase modulators PM1 and PM2 are 50%, the total propagation loss aL of the OFRR is 0.05 dB, the output power of the fiber laser is 1 mW, the gain of the demodulation circuit is 1, and the photoelectric transformation coefficient NPD is 6.25 V/mW.
ð9Þ
where Dy1 is the phase shift for one stair when the stair-period of the digital ramp phase waveform is t, i.e. p in Eq. (8) equals 1. Set the instant phase induced by the digital ramp phase waveform for Ethr(t) as j(t), if p ¼1, then j(t) can be written as
jðtÞ ¼ m Dy1
ð10Þ
where m is an integer. The corresponding phase induced by the digital ramp phase waveform for Encro ðtÞ can be written as
jðtntÞ ¼ ðmnÞDy1
ð11Þ
3.1. Influence of the output signal of the OFRR
If pa1, that is to say the stair-period is not equivalent to the transmission time in the OFRR then the phase induced by the digital ramp phase for Ethr(t) and Encro ðtÞ can be written as m þ1 m þ 1n jðtÞ ¼ Dyp , jðtntÞ ¼ Dyp ð12Þ p p int int
The digital ramp voltage applied with the phase modulator achieves the optical frequency shift. Phase of the lightwave is modulated by f(t) as shown in Fig. 2 before being launched into the OFRR. A general two-frequency combined digital serrodyne phase modulation waveform alternating between the two different slopes of F1 and F2 with the repetition rate of q is shown in Fig. 2(a). A special case for the symmetrically digital triangle phase waveform alternating between the positive and negative slopes with the repetition rate of F/2 is shown in Fig. 2(b); t0 is the stair period of the digital ramp phase waveform and Dy the phase shift for one stair. F is defined as the slope of the
where [ ]int is the largest integer smaller than the value in the bracket. When p is an integer greater than 1, the circulating fields achieve a phase shift of Dyp only after circulating in the OFRR for p turns, and p successive components share the same phase shift. For every cross field Encro ðtÞ circulated in the OFRR n turn, it has p possible phase shifts. Table 1 shows the phase shift introduced by the digital ramp waveform for Ethr(t) and Encro ðtÞ in the case of p¼1 and 2, respectively. As seen from Table 1, there are two possible phase shifts for the optical field Encro ðtÞ with n being an odd number. Fig. 3 shows the output signal of PD1 when the laser frequency f0 equals the resonant frequency fc in the case of p¼1, 2, 10, 1.1, 1.8, and 2.1. The frequency shift of the digital serrodyne phase
(t)
2π repetition frequency q
′ 1/F1
1/F2 t
(t)
2π
′ 1/F
′ 1/F
1/F
1/F
t Fig. 2. Digital ramp phase waveforms. (a) A general two-frequency combined digital serrodyne phase modulation waveform alternating between the two different slopes of F1 and F2 with the repetition rate of q and (b) a special case for the symmetrically digital triangle phase waveform alternating between the positive and negative slope with the repetition rate of F/2.
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Table 1 Phase shift introduced by the digital ramp phase waveform for Ethr(t) and Encro ðtÞ for the case of p ¼ 1 and 2, respectively. Optical field
j(t) phase shift induced by the digital ramp phase waveform and Dyp phase shift for one step
Phase shift
j(t)(p ¼ 1) ot+ j0 + j(t) ot+ j0 ot + j (t t) o + j0t 2ot + j (t 2t) ot+ j0 3ot + j (t 3t) ot+ j0 4ot + j (t 4t)
Ethr E1cro E2cro E3cro E4cro ^ Encro
j(t)(p ¼ 2) m Dy2
m Dy1 (m 1)Dy1 (m 2)Dy1 (m 3)Dy1 (m 4)Dy1 ^ (m n)Dy1
^
ot+ j0 not + j (t nt)
m Dy2 Dy2 m Dy2 Dy2 m Dy2 2Dy2 m Dy2 2Dy2 ^ nþ1 m Dy2 Dy2 2 int
m Dy2 m Dy2 Dy2 m Dy2 Dy2 m Dy2 2Dy2 h ni m Dy2 Dy2 2 int
m is an integer.
0.09
0.09
0.20
0.07 0.06
Intensity at PD1
Intensity at PD1
Intensity at PD1
0.08
0.08 0.07 0.06 0.05
0.05 0
1
2 3 t (μsec)
4
1
2 3 t (μsec)
4
0
1
2 3 t (μsec)
0.09
0.06 0.05
Intensity at PD1
Intensity at PD1
Intensity at PD1
0.07
0.08 0.07 0.06
5
4
5
p = 2.1
0.08 0.07 0.06 0.05
0.05 4
0.05
p = 1.8
0.08
2 3 t (μsec)
0.10
5
0.09 p = 1.1
1
0.15
0.00 0
5
0.09
0
p = 10
p=2
p=1
0
1
2 3 t (μsec)
4
5
0
1
2 3 t (μsec)
4
5
Fig. 3. Output signal of the FORR in the case of p ¼ 1, 2, 10, 1.1, 1.8, and 2.1.
waveform is 100 kHz. As seen from Fig. 3, the output signal of PD1 has a steady value with p¼1. In the case of p¼2 and 10, respectively, the output signal of PD1 is interchanging between two and ten different values. As p increases, the possible phase shifts of the cross-fields increase, so the outputs are interchanging between p different values. When the reciprocal of p is an integer greater than 1, the circulating fields achieve 1/p multiples of Dyp after circulating the FORR for one turn. The output of PD1 is kept at a steady value, the same as with the case of p¼ 1. When p or 1/p is not an integer, the phase shift of the circulating field is more complex. The output field fluctuates circulating around the ideal value with the period of t. The output values are sampled with the period of t; after averaging, there is still some deviation between the mean value and the ideal value. It can be further found that the deviation is dependent on p. When p is deviated further off one, the deviation of the output is bigger.
frequency of the lightwave launching into the OFRR, the resonance curve of OFRR is observed at PD. Fig. 4 shows the resonance curves with the frequency range of one free spectral range (FSR) in the case of p¼ 1, 2.5, and 10. Here FSR ¼c/nrL. As seen from Fig. 4, when p is an integer greater than 1, there are p periods duration of resonance curves in the frequency range of one FSR. It seems that the frequency distance between two neighboring resonance dips is compressed as 1/p of FSR. According to Eq. (7), when the phase shift of one stair of digital ramp phase waveform satisfies
Dyp
tu
¼
2p
t
the frequency shift induced by the digital ramp phase waveform is one FSR. Combining Eq. (8) and (13), Dyp satisfies
Dyp ¼ p2p 3.2. Influence of the resonance curve Setting the frequency of fiber laser equal to the resonant frequency of the OFRR and changing the slope of the digital ramp phase waveform linearly with time is used to sweep the
ð13Þ
ð14Þ
Sweeping the frequency of the digital ramp phase waveform for one FSR, the phase shift Dyp of one stair should equal 2pp. In fact, one resonance dip comes out for every phase shift of 2p. Pseudo-resonance dips come out when p is bigger than 1 as shown in Fig. 4. In order to avoid the influence of the
H. Ma et al. / Optics and Lasers in Engineering 48 (2010) 933–939
p=1 p = 2.5 p = 10
1
937
0.6 p = 20
0.9
Intensity at PD1
0.7 0.6 Iout
p = 10 p=1
0.5
0.8
0.5 0.4
0.4 0.3 0.2
0.3
0.1
0.2 0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 frequency of digital serrodyne waveform *FSR
0.0 -300
-200
-100
0.5
0 100 frequency (kHz)
300
0.6
Fig. 4. Resonance curves of the OFRR in the case of p ¼ 1, 2.5, and 10.
p = 20 p = 10 p=1
0.5
Intensity at PD1
p=1 p = 0.8 p = 0.5
1
200
0.9 0.8 0.7
0.4 0.3 0.2
Iout
0.6 0.1
0.5 0.4
0.0 -400
0.3
-300
-200
0 -100 frequency (kHz)
100
200
0.2 0.1
0.6 p = 20 p = 10 p=1
0 1
1.5
frequency of digital serrodyne waveform
-1.5
-1
-0.5
0
0.5
*FSR
2
0.5
Fig. 5. Resonance curves for the frequency range of four FSR in the case of p ¼1, 0.8, and 0.5.
0.4
pseudo-resonance dips, the slope of the digital ramp phase waveform should be lower than 2p FSR/p. Fig. 5 shows the resonance curves for the frequency range of four FSRs in the case of p¼1, 0.8, and 0.5. When p is less than 1, the frequency distance between two neighboring resonance dips is elongated 1/p times of FSR. Only one cycle of the resonance curve appears in 1/p times of FSR range. Next we will discuss the influence of the position of the resonance dip. Fig. 6 shows the resonance curves around the resonance dip while sweeping the frequency shift of the digital ramp phase waveform in the case of p ¼1, 10, and 20, respectively. Fig. 6(a) shows the resonance curve for f0 ¼fc. Fig. 6(b) shows the results for f0 ¼fc +100 kHz and Fig. 6(c) shows the results for f0 ¼fc 100 kHz. Solid line for p ¼20, dash line is for p ¼10, and dot line for p¼ 1. As seen from Fig. 6, for f0 ¼ fc, the resonance curves are apart for different values of p but they have the same resonance dips. For f0 ¼fc + 100 kHz, the resonance curves are upshifted and the resonance dip has moved rightwards as p
Intensity at PD1
-2
0.3 0.2 0.1 0.0 -200
-100
0
200 100 frequency (kHz)
300
400
Fig. 6. Resonance curves around the resonance dip in the case of p ¼1, 10, and 20. (a) f0 ¼fc; (b) f0 ¼ fc + 100 kHz; and (c) f0 ¼fc 100 kHz.
increases. For f0 ¼ fc 100 kHz, the resonance curves are upshifted too; however the resonance dip has moved leftwards as p increases. The movement of the resonance dip affects the lockin position of the center frequency of the laser. This means that the laser frequency is not locked to the practical resonant
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frequency, which affects the rotation rate readout directly. Through analyzing the simulation results in Fig. 6, it can be found that there is no deviation of the resonance dip when the laser frequency is the same as the resonant frequency. That is to say, if the lock-in state corresponds to f0 ¼fc, the laser frequency is locked to the practical resonant point in spite of any values of p. The error in the resonant frequency (Dfr) is dependent on p and the difference between the laser frequency and the resonant frequency (Df¼f0 fc). Fig. 7 shows the relationship between the error in the resonant frequency and the resonant frequency deviation Df. Two cases with p of 0.8 and 2 are calculated. Dividing Dfr by Df introduces a unit error in the resonant frequency Dfr0. Then Dfr can be written as
Dfr ¼ Dfr0 Df
ð15Þ
Eq. (15) indicates that when Df is 0, Dfr is kept at zero in spite of any values of Dfr0. Fig. 8 shows the relationship between the unit resonant frequency deviation Dfr0 and p. As seen from Fig. 8, there is no error for the ideal stair period (p ¼1). Errors appear for both decimal and integer numbers of p expect for the ideal stair period (p ¼1) as shown in Fig. 8. When p is closer to 1, the resonant frequency deviation is smaller.
3.3. Zero-deviation of the demodulation curve Previous analysis is done only for a single digital ramp phase waveform applied with the phase modulator in the RFOG. A combined waveform with two different slopes is used to detect the gyro signal in the practical RFOG. For a two-frequency combined serrodyne modulation waveform [12], the twofrequency shifts are F1 and F2. For the symmetric triangle waveform, the two-frequency shifts are F and F. Three cases of p¼ 1, p ¼10, and p ¼20 are calculated. Fig. 9 shows the demodulation curves for the digital triangle phase waveform with the frequency shifts 80 and 80 kHz applied with the phase modulator. The lock-in state is f0 + (80 80) kHz/2¼fc, according to Eq. (15), Dfr is always zero in the case of Df¼ 0 in spite of any values of p. So the laser frequency is locked to the resonant frequency and the demodulation output is zero. There is no zero-deviation of the demodulation curve regardless of any values of p. The same result is shown in Fig. 9. Though there is some difference in the amplitude of the resonance curves as shown in Fig. 6, amplitudes of the three demodulation curves are very similar. The good symmetry in the resonance curves around the resonance dip counteracts the difference in the amplitude of the
0.40
p=1 p = 10 p = 20
0.35
p=2
0.2 0.15
0.30 demodulation value
error in the resonant frequency /Hz
0.25
0.25 0.20 0.15 0.10
0.1 0.05 0 -0.05 -0.1 -0.15
p = 0.8
-0.2
0.05
-0.25
0.00 40
60
80
100 f0-fc/Hz
140
120
-0.05 -0.04 -0.03 -0.02 -0.01
0
frequency deviation
Fig. 7. Relationship between the error in the resonant frequency (Dfr) and the resonance frequency deviation (f0 fc) in the case of p ¼ 0.8 and 2, respectively.
*FSR
Fig. 9. Demodulation curve for the symmetrically digital triangle phase waveform with frequency shifts 80 and 80 kHz.
0.1
0.0030
p=1 p = 10 p = 20
0.08
0.0025
0.06 demodulation value
error in the unit resonance frequency
0.01 0.02 0.03 0.04 0.05
0.0020 0.0015 0.0010 0.0005
0.04
6kHz
0.02 22kHz
0 -0.02 -0.04 -0.06 -0.08
0.0000 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
p Fig. 8. Relationship between the error in the unit resonance frequency (Dfr0) with p.
-0.1 -0.05 -0.04 -0.03 -0.02 -0.01
0
frequency deviation
0.01 0.02 0.03 0.04 0.05 *FSR
Fig. 10. Demodulation curve for the two-frequency combined digital serrodyne phase waveform with frequency shifts 50 and 100 kHz.
H. Ma et al. / Optics and Lasers in Engineering 48 (2010) 933–939
demodulation curves. Fig. 10 shows the demodulation curve for the two-frequency combined digital serrodyne phase modulation waveform with the frequency shifts 50 and 100 kHz applied with the phase modulator. The locked condition is f0 + (50+ 100) k/2 ¼fc, i.e Df ¼ 75 kHz, so there is a zero-deviation of the demodulation curve in the case of p a1. The zero-deviation is about 6 and 22 kHz in the case of p¼10 and p¼20, respectively. For an OFRR with a diameter of 10 cm, the gyro scale factor is about 44 kHz/ (rad/s). This big zero-deviation needs to be avoided.
4. Conclusion Errors in RFOG based on the digital ramp phase modulation technique are analyzed. The physical mechanism for the error is the uncertain phase difference between the transmitted light and circulated light components in the OFRR. The zero-deviation of the demodulation curves in the symmetrically digital triangle and the two-frequency combined serrodyne phase modulation waveforms are compared. It is found that the zero-position of the demodulation curve in the symmetrically digital triangle phase modulation technique is not affected by the nonideal stair-period problem. In other words, the symmetrically digital triangle wave phase modulation technique is more advantageous in reducing the nonideal stair-period problem. The theoretical results are helpful to optimize the parameters in the digitalized RFOG system.
Acknowledgement The authors thank the Specialized Research Fund for the Doctoral Program of High Education, PR China (20060335064) for financial support.
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