Alternative method for design and optimization of the ring resonator used in micro-optic gyro Kunbo Wang,1,2,* Lishuang Feng,1,2 Junjie Wang,1,2 and Ming Lei1,2 1
Key Laboratory on Inertial Science and Technology, Beihang University, Beijing 100191, China
2
Key Laboratory of Micro-nano Measurement-Manipulation and Physics (Ministry of Education), Beihang University, Beijing 100191, China *Corresponding author:
[email protected] Received 18 October 2012; revised 21 December 2012; accepted 23 January 2013; posted 31 January 2013 (Doc. ID 178278); published 28 February 2013
The ring resonator is one of the key elements in the micro-optic gyro system, but there is not a uniform method for designing the parameters of a ring resonator, especially for its size. In this paper, an alternative method is presented for designing the ring resonator used in micro-optic gyro. Maximization of the resonator output is proposed to be the principle in design and optimization for the first time to our knowledge. The scale factor accuracy and the full range of the gyro system are taken into account to obtain the optimum diameter of the ring. A theoretical optimal diameter of 0.25 m is achieved for SiO2 waveguide resonator with a dynamic range of 500°∕s by analyzing the influence of resonator parameters on the output in detail, and the corresponding sensitivity of the gyro is less than 1.28°∕h, which can meet the demands of a tactical inertia system. © 2013 Optical Society of America OCIS codes: 060.2800, 130.6010, 230.4555.
1. Introduction
Resonant micro-optic gyro (RMOG), which has excellent advantages such as high reliability, wide dynamic range, small size, and low cost, is one of the most promising candidates for the next-generation inertial rotation sensors [1–3]. Angular velocity can be obtained in RMOG by sensing the resonant frequency difference between the counterpropagating light beams transmitting in the waveguide ring resonator. The ring resonator is one of the key elements in resonant gyro. Most researchers optimize parameters of the resonator to obtain high-performance sensors. Shupe [4] presented the transfer function for transmitting a fiber resonator and analyzed the sensitivity (δΩ) of gyro in detail, pointing out that there existed an optimal length of fiber for the highest sensitivity and the corresponding coupling coefficient increases as the length of the fiber increases. Sanders et al. [5] showed a different description of δΩ and only depicted 1559-128X/13/071481-06$15.00/0 © 2013 Optical Society of America
the influence of integration time. Iwatsuki et al. [6] presented another expression of δΩ, where the optical source spectrum width was taken into account, and the impact of resonator length on sensitivity was also emphasized. All the three optimizations only considered the sensitivity of gyro, while other aspects such as scale factor accuracy and full range of gyro were not mentioned. In this paper, we propose an alternative procedure to design the ring resonator. Maximization of the resonator output is proposed to be the principle in the design and optimization. The resonator input/output characteristics are analyzed and the relationship between the resonator output and resonator parameters is achieved in Section 2. In Section 3, numerical simulations are performed with respect to the SiO2 waveguide and the optimal diameter for the maximum output of the resonator is obtained for the first time to our knowledge. The scale factor accuracy and the full range of gyro are considered to satisfy the tactical grade system. After comparing it with the traditional methods, the method proposed is proved to be effective in design of the waveguide resonator. 1 March 2013 / Vol. 52, No. 7 / APPLIED OPTICS
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8 p T 1 − κ1 − αc > > > p > 0 > > − αc 1 − αL < R κ1 p Q0 1 − αL 1 − κ1 − αc ; > > > > R R0 exp−πΔf 0 τ > > : Q Q0 exp−πΔf 0 τ Fig. 1. (Color online) Common configuration of ring resonator used in RMOG. D, diameter; κ, cross-port coupling coefficient of the resonator coupler; Iin , input light intensity; Iout , output light intensity. Here, D and κ are the major two parameters that influence the characteristics of the resonator.
2. Theoretical Framework
The common configuration of the ring resonator is shown in Fig. 1. The resonant frequency difference Δf between the clockwise (CW) and the counterclockwise (CCW) beams propagating in the optical ring resonator according to the Sagnac effect is given by [7]
Δf
4A Ω; neff λL
(1)
where A is the area enclosed by the resonator, L is the optical perimeter, neff is the waveguide refractive index, λ is the wavelength of the light, and Ω is the rotation rate. Note that Δf is independent of the number of turns in the waveguide resonator. Considering the temporal coherence of the laser, the resonance curve of a ring resonator is expressed as [8,9]
T frr
I out 2TR cos2πτf − M T2 − ; I in 1 Q2 − 2Q cos2πτf
(2)
where M 2TRQ
R0 2 · 1 − Q2 ; 1 − Q0 2
(3)
4
where αc and αL are the loss of the resonator coupler and the resonator, respectively. Δf 0 is the spectral linewidth of the laser, f is the input light frequency, and τ is the optical transmission time in the ring expressed as τ neff · L∕c. Here c is the velocity of light. The transfer function expressed by Eq. (2) has been plotted [Fig. 2(a)] with parameters in Table 1. As can be seen from Fig. 2(a), ΔT frr is different for the same Δf at different working frequency and it is almost proportional to the slope of the resonance curve. In RMOG, high performance corresponds to high signalto-noise ratio (SNR), which demands larger ΔT frr when I in is fixed. Therefore, it is necessary to shift the frequency to the maximum slope point of the resonance curve in order to obtain the largest ΔT frr . The slope K of the resonance curve by fitting Eq. (5) (n 1; 2; 3…) is shown in Fig. 2(b), whose absolute value is symmetrically distributed as the resonance curve. For the maximum jKj, K max , the corresponding frequency f m 9.66 MHz is obtained, which would be assumed to be the working frequency for higher performance. In order to confirm that the scale factor accuracy of gyro can meet the demand for tactical application, the nonlinearity of the range −f m 1 MHz in the resonance curve is calculated by Eq. (6), as shown in Fig. 2(c): dT frr T f − T frr f n−1 frr n df f n f n − f n−1 M − 2TR cos2πτf n 1 Q2 − 2Q cos2πτf n M − 2TR cos2πτf n−1 ∕f n − f n−1 ; (5) − 1 Q2 − 2Q cos2πτf n−1
K
Fig. 2. (Color online) (a) Resonance curve of resonator with parameters of Table 1. FWHM is the full width at half maximum. (b) Slope of the resonance curve. (c) Nonlinearity of the resonance curve. 1482
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Table 1.
Parameters for the SiO2 ∕SiO2 :Ge∕SiO2 Waveguide Resonator
Characteristics Wavelength of light Linewidth of laser Refractive index Transmitting loss per unit length Inserting loss of coupler Ring diameter Cross-port coupling coefficient
σ
Varies
Values
λ Δf 0 neff α αc D κ
1.55 μm 30 KHz 1.46 0.01 dB∕cm 0.3 dB 0.04 m 0.03
jjKj − max jKjj : max jKj
(6)
Figure 2(c) clearly shows that, in the range of −f m 0.35 MHz, the nonlinearity of the slope is less than 1000 ppm, which can satisfy the tactical grade system [10]. The corresponding angular velocity is 1135°∕s, and it will be assumed to be the full range of RMOG. In that case, the output intensity difference ΔI of the resonator can be described as ΔI I in · ΔT frr ≈ I in · K max · Δf :
(7)
For the round shape ring resonator, ΔI can be expressed as ΔI ≈ I in ·
K max D · Ω: neff λ
(8)
We define a parameter ΔI 0, which is independent of Ω and I in , to represent the performance of resonator, expressed as Eq. (9). The full range of gyro, Ωfull , should be determined by σ 1000 ppm to satisfy the tactical grade system [10]: ΔI 0
ΔI K D ≈ max ; I in · Ωfull neff λ
Ωfull Ωjσ1000 ppm :
As can be seen from Eq. (9), ΔI 0 is proportional to the multiple of K max and D. When the diameter D is fixed, we need to set the frequency bias at f m , where K max can be obtained to achieve higher ΔI 0. Different modulation techniques [11–13] can realize a frequency shift of f m. Figure 3(a) shows the square wave frequency modulation, which is much more concise than sine-wave modulation. When the gyro is stable, Δf 0, the modulated lights with frequencies of Δf − f m and Δf f m are symmetrically settled on two sides of the resonant curve, in which case ΔI 0 0. When the gyro rotates, CCW departs from CW, resulting in modulated lights’ frequencies asymmetric on the resonant curve and square-wave output, in which case ΔI 0 can be expressed exactly as in Eq. (11) by considering Eqs. (2) and (9), where Δf and the symbol “” are decided by Ωfull and the rotating direction, respectively. The normalized ΔI 0 obtained by Eq. (11) for different f m is shown in Fig. 3(b), and the derivative K of the resonant curve is also listed for comparison. As it can be seen, ΔI 0 has the same trend with K for different frequency when the resonator parameters have been fixed, and it reaches largest at the point of f m 9.66 MHz, where the slope of resonant curve K is maximum. This trend confirms that the resonator parameters, such as diameter and the cross-port coupling coefficient, should be optimized to achieve bigger K for higher performance of RMOG: I in · T frr jΔf −f m − T frr jΔf f m ΔI I in · Ωfull I in · Ωfull 1 2TR cos2πτΔf f m M · Ωfull 1 Q2 − 2Q cos2πτΔf f m 2TR cos2πτΔf − f m M : − 1 Q2 − 2Q cos2πτΔf − f m
ΔI 0
(11)
3. Theoretical Results and Discussion
(9) (10)
From the analysis in the previous section, we conclude that the resonator structure parameters, such as diameter D and cross-port coupling coefficient κ, should be optimized to realize high-performance sensors.
Fig. 3. (Color online) (a) Modulation technique of the RMOG. (b) Influence of modulation frequency f m on ΔI0 . (The calculated parameters are listed in Table 1.) 1 March 2013 / Vol. 52, No. 7 / APPLIED OPTICS
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Fig. 4. (Color online) Influence of D and κ on the slope of resonance curve K. (a) Slope of resonance curve with different κD 0.04 m. (b) Relationship between K max and κD 0.04 m. (c) Slope of resonance curve with different Dκ 0.05. (d) Relationship between K max and D, where for each point, κ has been optimized for maximum K.
A.
Relationship between K max and D, κ
Figure 4 shows the influence of D and κ on the slope of the resonance curve, where FSR is the free spectrum range of the resonance curve. As can be observed, there exist optimum D and κ for maximum K max . B.
Relationship between
ΔI 0
Comparison and Discussion
In order to illustrate the validity of the method, we calculate three different descriptions of gyro 1484
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1000 D = 3.93m
0.8
and D
The relationship between ΔI 0 and D is calculated using the parameters in Table 1. Considering that D has a larger range versus to κ0 ∼ 1, the procedure of the optimal κ is contained in the variation of D. By plotting ΔI 0 with different values of D, optimal diameter of the resonator is achieved [Fig. 5(a)]. For the maximum ΔI 0 , D is 3.93 m, while the corresponding full range of gyro is too tiny to fulfill the tactical grade performance; therefore, D should be less than 0.25 m to make the full range larger than 500°∕s [10] [Fig. 5(b)], in which case κ equals 0.14. The resonance curve with diameter of 0.25 m is shown in Fig. 5(c), and the characteristics of the resonator are illustrated in Table 2, where F is finesse, described as the ratio between FSR and FWHM; ρ is the resonance depth, defined as the ratio between the amplitude range and the maximum value of the resonance curve. C.
1.0
0.6
600
0.4
400
Full range: 500°/s
0.2 0.0
800
200
0
1
2 3 Diameter D (m)
4
5
0
(a) 1.0 0.8
0.10
0.6 FSR: 261MHz
0.4
0.05
0.2 0
0
0.1 0.2 0.3 0.4 Diameter D (m)
0.0 -0.5
(b)
0.0
0.5 FSR
1.0
1.5
(c) ΔI 0
Fig. 5. (Color online) (a) Calculated (solid curve) and full range of RMOG (blue dashed curve) versus D. Dashed horizontal line: full range is 500°∕s. (b) Detailed distribution of ΔI 0 near the full range of 500°∕s. (c) Resonance curve of the optimum resonator with D 0.25 m and κ 0.14.
Table 2.
Characteristics of the Optimum Resonator and the Resonator in Reference
δΩ °∕h Dm (m) 0.25 [this paper] 0.035 [11]
κ
FSR (GHz)
F
ρ
Eq. (12)
Eq. (13)
Eq. (14)
0.14 0.03
0.261 1.61
13.4 50
0.79 0.74
1.28 21.71
0.69 10.12
0.69 13.23
Fig. 6. (Color online) Calculated ΔI 0 (solid curve) and sensitivity (dashed curves) versus (a) diameter D and (b) cross-port coupling coefficient κ for D 0.04 m. A, B, and C are for Eqs. (12)–(14), respectively.
sensitivity δΩ presented in [4–6], which are respectively given by p 2Γ λL ; · δΩ ≈ 4A SNR
(12)
p λL 2Γ · ; δΩ ≈ 4A N ph ηt1∕2
(13)
p p ℏω 1∕2 3 3 Ncλ π· Γ ; 1 − 2∕3ρ δΩ 4A ηI 0 FρL2
(14)
where Γ is FWHM, and η 0.72 is the detector quantum efficiency. Other variables are described as in each reference and some initial values are assumed as in Table 1. The influence of D and κ on δΩ is reported (Fig. 6) with Eqs. (12)–(14). As can be seen, the δΩ values of three formulas are different at the same diameter, but they are of the same magnitude order. The reason may be that the three formulas have different simplification. The diameters for minimum δΩ are almost the same in Eqs. (12) and (14) as in our study [Fig. 6(a)], and the optimum κ is similar to the value obtained by our method when D 0.04 m [Fig. 6(b)]. The three approaches only consider the sensitivity of gyro, while the full range and scale factor accuracy are not mentioned. Three values of sensitivity, calculated by Eqs. (12)–(14) with the optimum resonator in this paper and that in [11], are shown in Table 2. Obviously, RMOG based on the resonator optimized using this method has higher sensitivity; however, it is too large
to fabricate, and two optimum approaches are proposed for different resonator configurations. When using multiturn, we should first calculate the maximum slope K max to get the diameter Dkmax and the coupling coefficient κ kmax by Eq. (5). Choose κkmax to be the cross-port coupling coefficient and Dkmax ∕p to be the resonator diameter, where p is an integer. When using a single turn, we should confirm diameter D first, and then calculate the coupling coefficient κ to derive K max ; therefore D and κ are the optimum parameters. It can be concluded that the method proposed in this paper shows an obvious rule for the design of the resonator used in RMOG, and the diameter in practice should be equal to or smaller than the theoretical value to realize higherperformance sensors. 4. Conclusions
In summary, a new design method is presented to achieve the optimum parameters of the ring resonator used in RMOG. The maximization of the resonator output is proposed to be the principle in designing and optimizing the resonator for the first time to our knowledge with the scale factor accuracy and the full range of the gyro system both taken into account. In the end, the optimum diameter and cross-port coupling coefficient are obtained to be 0.25 and 0.14 m, respectively, with respect to the SiO2 ∕SiO2 :Ge∕SiO2 waveguide. After comparing with resonator parameters used in other references, the optimized values in this paper result in higher sensitivity of gyro. Although the optimum diameter of the resonator is too large to fabricate, it shows a maximum value, and the diameter in practice should be smaller than that. This method will provide better guidance for the design of the resonator used in RMOG. 1 March 2013 / Vol. 52, No. 7 / APPLIED OPTICS
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The authors thank the National Natural Science Foundation of China under grant No. 61171004 for financial support. References 1. K. Suzuki, K. Takiguchi, and K. Hotate, “Monolithically integrated resonator microoptic gyro on silica planar lightwave circuit,” J. Lightwave Technol. 18, 66–72 (2000). 2. H. Ma, X. Zhang, Z. Jin, and C. Ding, “Waveguide-type optical passive ring resonator gyro using phase modulation spectroscopy technique,” Opt. Eng. 45, 80506 (2006). 3. H. Ma, Z. He, and K. Hotate, “Reduction of backscattering induced noise by carrier suppression in waveguide-type optical ring resonator gyro,” J. Lightwave Technol. 29, 85–90 (2011). 4. D. M. Shupe, “Fiber resonator gyroscope: sensitivity and thermal nonreciprocity,” Appl. Opt. 20, 286–289 (1981). 5. G. A. Sanders, M. G. Prentiss, and S. Ezekiel, “Passive ring resonator method for sensitivity inertial rotation measurements in geophysics and relativity,” Opt. Lett. 6, 569–571 (1981).
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6. K. Iwatsuki, K. Hotate, and M. Higashiguchi, “Effect of Rayleigh backscattering in an optical passive ring-resonator gyro,” Appl. Opt. 23, 3916–3924 (1984). 7. A. H. Rosenthal, “Regenerative circulatory multiple-beam interferometry for the study of light-propagation effects,” J. Opt. Soc. Am. 52, 1143–1147 (1962). 8. Y. Ohtsuka, “Analysis of a fiber-optic passive, loop-resonator gyroscope: dependence on resonator parameters and lightsource coherence,” J. Lightwave Technol. LT-3, 378–384 (1985). 9. H. Ma, Z. Jin, C. Ding, and Y. Wang, “Influence of spectral linewidth of laser on resonance characteristics in fiber ring resonator,” Chin. J. Lasers 30, 731–734 (2003). 10. A. Lawrence, Modern Inertial Technology: Navigation, Guidance, and Control (Springer, 1998). 11. H. Yu, C. Zhang, L. Feng, L. Hong, and M. Lei, “Limitation of rotation sensing in IORG by Rayleigh backscattering noise,” Eur. Phys. Lett. 95, 64001 (2011). 12. X. Zhang, H. Ma, Z. Jin, and C. Ding, “Open-loop operation experiments in a resonator fiber-optic gyro using the phase modulation spectroscopy technique,” Appl. Opt. 45, 7961–7965 (2006). 13. H. Mao, H. Ma, and Z. Jin, “Polarization maintaining silica waveguide resonator optic gyro using double phase modulation technique,” Opt. Express 19, 4632–4643 (2011).
