Effect of Fresnel reflections in a hybrid air-core photonic-bandgap fiber ring-resonator gyro

Yuchao Yan; Huilian Ma; Linglan Wang; Hanzhao Li; Zhonghe Jin

doi:10.1364/OE.23.031384

1. Introduction

The resonant fiber optic gyro (RFOG) is a solid state inertial rotation rate sensor based on the Sagnac effect [1]. Compared with the interferometric fiber optic gyro (IFOG), the RFOG has the potential to achieve the same performance with coil lengths of up to 100 times shorter than that of the IFOG benefiting from the resonance effect [2–4]. However, under the investigation for years in laboratory, the bias stability of the RFOG is still below expectation. Several drift mechanisms, such as polarization fluctuation [5–7] and optical Kerr effect [8,9], were reported as the main barriers to the performance improvement. The emergence of the air-core photonic bandgap fiber (PBF) offers a novel approach for the development of the RFOG. Unlike the solid-core polarization-maintaining (PM) fiber, the most energy of the lightwave in the air-core PBF is confined by the air-guide mechanism [10]. Simulations and experiments indicate that the optical Kerr coefficient is about 250 times smaller than in a Corning′s SMF-28 fiber [11], and the thermal polarization instability is 270 times lower than in a conventional solid-core PM fiber [12]. It can be expected that the air-core PBFs based RFOG should benefit from those advantages as well.

A high-sensitivity RFOG requires a low-loss optical fiber ring resonator. Unfortunately, there is no commercial low-loss air-core PBF optical coupler being able to construct a low-loss air-core PBF ring resonator to date. Many groups have proposed different configurations of air-core PBFs incorporated into the resonator [13–16]. Made by free-space coupling with bulk optics, the measured finesse is approximately 42, which corresponds to a round-trip loss of only 7% [13]. However, the low aseismic capacity of the free-space coupling structure confines the gyro from practical application. Feng et al. proposed a new type of coupling structure based on a micro-optical splitter. The measured resonant finesse is 11.75, however, the resonant depth is only 0.087 due to the actual coupling loss much greater than its theoretical counterpart [15]. There is no further RFOG result demonstrated in the aforementioned two resonators. An alternative approach was proposed to construct a hybrid air-core PBF ring resonator by using a conventional solid-core single-mode fiber (SMF) optical coupler [14]. The long-term drift with a standard deviation of about 0.5°/s was observed in this hybrid air-core PBF ring-resonator gyro [14]. Due to the high splice loss and large Fresnel reflections at the junction between the air-core PBF and the pigtails of the SMF coupler, as well as the large polarization fluctuation from the SMF coupler, the achieved bias stability is almost three orders of magnitude worse than those conventional RFOGs equipped with solid-core PM fiber ring resonators [17].

For hybrid air-core PBF ring resonator applications, it is critical to be able to splice air-core PBFs to conventional solid-core fibers with low loss. A lot of efforts have been done to reduce the splice loss and the lowest data reported to date is 1.3 dB per junction [18]. In addition to loss reduction, low reflection and high polarization extinction ratio (PER) are also indispensible for a high-performance RFOG.

This article proposes a hybrid PM air-core PBF ring resonator by splicing a section of PM air-core PBF to a solid-core PM fiber optical coupler. The utilization of the PM fiber optical coupler is able to keep the high PER stability of the hybrid ring resonator. Nonetheless, both high splice losses and Fresnel reflections exist at the junctions between air-core PBFs and solid-core fibers. The forward output signal of the hybrid ring resonator is the normal resonant curve with the superposition of the lightwaves that experienced even numbers of Fresnel reflections. The backward output signal of the hybrid ring resonator is composed of lightwaves that experienced odd numbers of Fresnel reflections. Rigorous derivations are then provided. The biggest resonant depth and finesse of the hybrid air-core PBF ring resonator predicted are 0.352 and 6.3 respectively, by assuming a splice loss of 1.8 dB per junction. These predictions are finally confirmed by testing the output signals of the hybrid ring resonator. By applying this hybrid PM air-core PBF ring resonator to the RFOG, a bias stability of 0.007°/s is successfully demonstrated for operation times of 3600 seconds, which is the best bias stability reported to date in a hybrid air-core PBF ring-resonator gyro to our knowledge.

2. Principle and analysis

The schematic diagram of the input, transmitted, and reflected waves at an interface between the conventional solid-core PM fiber and the air-core PBF is shown in Fig. 1. The mode-field diameter of the PM fiber is slightly larger than that of the air-core PBF. Figure 1(a) shows the incident lightwave from the PM fiber to the air-core PBF, and Fig. 1(b) shows the incident lightwave from the air-core PBF to the PM fiber. The refractive index of the PM fiber is n_PM = 1.455, and the refractive index of the air-core PBF is n_PBF = 1.

Fig. 1 Fresnel’s laws at the interface between the air-core PBF and the conventional soild-core PM fiber.

Download Full Size | PDF

The fraction of the incident optical intensity that is reflected from the interface is defined as the reflectance R, and the fraction that is transmitted is given by the transmittance T. According to the Fresnel's laws, the reflection coefficient R at the junction between these two fibers are given by

R = 10^{- α_{s} / 10} \cdot {(\frac{n_{P M} - n_{P B F}}{n_{P M} + n_{P B F}})}^{2} = 0.02370

where α_S is the splice loss at the junction between the air-core PBF and the solid-core fiber. It is assumed to be 1.8 dB per junction, which can be verified by further experiments. The transmission coefficient T at the junction between these two fibers are given by

T = 10^{- α_{s} / 10} = 0.6673.

Every time the lightwave passes though the splicing point while propagating in the hybrid ring resonator, the optical intensity accounted for a percentage of R of the total intensity reflects and transmits for a percentage of T in the opposite direction. Because n_PM>n_PBF, while the lightwave inputs to the PBF from the PM fiber, the incident wave and the reflected wave are in the same direction and have no phase change. However, while the lightwave inputs to the PM fiber from the PBF, a phase change of π occurs in the transmitted wave. The phase of the reflected wave experiences no sudden phase change in all the cases.

Figure 2 shows the configuration of the hybrid air-core PBF ring resonator. P₁ and P₂ denote the two splicing points between the air-core PBF and the pigtails of the solid-core PM fiber coupler. We consider the lightwave propagating in the clockwise (CW) direction as an example. When the lightwave passes through the splicing point P₂, the transmitted wave continues to propagate in the CW direction while the reflected wave is propagating in the opposite direction, namely in the counterclockwise (CCW) direction. Both the transmitted and reflected waves will pass through the splicing point P₁ where the transmission and reflection effects happen again. Those multiple Fresnel reflections form an effect similar with a couple-resonator [19]. The forward output signal is the normal resonant curve with the superposition of the lightwaves that experienced even numbers of Fresnel reflections. The backward output signal is composed of lightwaves that experienced odd numbers of Fresnel reflections. Considering the phase change of π occurs while the lightwave inputs to the PM fiber from the PBF, the sign of the reflected wave changes as shown in Fig. 2.

Fig. 2 Configuration of the hybrid air-core PBF ring resonator.

Download Full Size | PDF

Figure 3 shows the mathematical model used to analyze the transfer functions of this hybrid air-core PBF ring resonator. K and α_c are the coupling coefficient and the coupling loss of the PM fiber optical coupler, respectively. a_PM, a_PBF and τ₁, τ₂ are the total propagation losses and transit times of the solid-core PM fiber and air-core PBF, respectively. ω is the central frequency of the lightwave.

Fig. 3 Mathematic model of the lightwave propagating in the hybrid air-core PBF ring resonator.

Download Full Size | PDF

In consideration of a 90° phase shift in the lightwave caused by the directional coupler, a phase factor i is multiplied. As shown in Fig. 3, the point P₁ indicates the lightwave input to the PBF from the PM fiber, while the point P₂ indicates the lightwave input to the PM fiber from the PBF. The resonant effect caused by the two splicing points P₁ and P₂ is shown in a dotted box. By introducing two representative parameters P and Q, a simplified version of the mathematic model is shown in Fig. 4

Fig. 4 Simplified mathematical model of the lightwave propagating in the hybrid air-core PBF ring resonator.

Download Full Size | PDF

The two representative parameters P and Q are expressed as

P = \frac{i \sqrt{K_{1} α_{c} a_{P M}} \exp i ω τ_{1} \cdot (1 - R a_{P B F} \exp 2 i ω τ_{2})}{1 - R a_{P B F} \exp 2 i ω τ_{2} - T a_{P M} \cdot \sqrt{(1 - K) α_{c} a_{P B F}} \cdot \exp i ω (2 τ_{1} + τ_{2})} .

Q = \frac{a_{P M} \sqrt{(1 - K) α_{c}} \cdot \exp 2 i ω τ_{1} (\sqrt{R} - \sqrt{R} (R + T) a_{P B F} \exp 2 i ω τ_{2})}{1 - R a_{P B F} \exp 2 i ω τ_{2} - T \cdot a_{P M} \sqrt{(1 - K) α_{c} a_{P B F}} \cdot \exp i ω (2 τ_{1} + τ_{2})} .

The final signal flow diagram is simplified as Fig. 5.

Fig. 5 Final signal flow graph of the transfer function.

Download Full Size | PDF

Let the function F(X) be expressed as

F (X) = \frac{T α_{P M} \sqrt{α_{P B F}} \exp i ω (X + 2 τ_{1})}{1 - R α_{P B F} \exp 2 i ω τ_{2}} .

And let G be expressed as

G = \sqrt{(1 - K) α_{c}} .

The transfer function H₁ and H₂ can be expressed as

H_{1} = G - \frac{K α_{c} ((- \sqrt{a_{P B F} R} F (2 τ_{2}) + a_{P M} \sqrt{R} \exp 2 i ω τ_{1}) \cdot Q + F (τ_{2}))}{1 - (a_{P M} \sqrt{R} \cdot G \cdot \exp 2 i ω τ_{1} - \sqrt{a_{P B F} R} G \cdot F (2 τ_{2})) \cdot Q - G \cdot F (τ_{2})} .

H_{2} = \frac{i \sqrt{K α_{c}} \cdot (- \frac{\sqrt{a_{P B F} R}}{\sqrt{a_{P M}}} F (2 τ_{2} - τ_{1}) + \sqrt{a_{P M} R} \exp i ω τ_{1}) \cdot P}{1 - (a_{P M} \exp 2 i ω τ_{1} - \sqrt{a_{P B F}} F (2 τ_{2})) \sqrt{R} \cdot G \cdot Q - G \cdot F (τ_{2})} .

According to Eq. (7) and Eq. (8), Fig. 6 shows the output signals of the hybrid air-core PBF ring resonator. The parameters used in the calculation are: K = 0.1, α_c = 0.34 dB, a_PM = 2 dB/km, and a_PBF = 28 dB/km. The total length of the PM fiber within the hybrid ring resonator is 2.4 m, the length of the air-core PBF is 10.5 m, and the splice loss is assumed to be 1.8 dB per junction. Compared with the normal output resonant curve, the period of the forward output signal is almost fixed. However, the even numbers of Fresnel reflections superimposing on the normal resonant curve result in the fluctuation of the forward output signal valley. Differing from the traditional Fabry–Pérot interferometer made by the two in-line mirrors of a certain period, the multiple Fresnel reflections at the two junctions propagating along different optical paths in the hybrid ring resonator cause the different periods of the resonant effects. Those different periods of the lightwaves that experienced even numbers of reflections deteriorate the stability of the resonant curve. The simulation result shows that affected by the even numbers of reflections, the biggest resonance depth of the hybrid ring resonator is approximately 0.352 and the finesse is about 6.3. After the lightwave reflects for odd times through the resonator, the resonant curve can be obtained and the fluctuation of whose peak value is more drastic. The maximums of the two output amplitudes are 38 times different from each other. Figure 6 also shows that when the superposition reflections of even times gets to maximum, the reflections of odd times gets to minimum, and vice versa.

Fig. 6 Simulation results of the output signals of the hybrid air-core PBF ring resonator.

Download Full Size | PDF

3. Experiment and result

The schematic for measurement of output signals of the hybrid PM air-core ring resonator is depicted in Fig. 7. A sawtooth-wave signal from a signal generator tunes the central frequency of a fiber laser (FL) linearly with respect to time. The forward output signal is the normal resonant curve with the superposition of lightwaves that experienced even numbers of Fresnel reflections, which is detected by photodetector PD2. The backward output signal is composed of lightwaves that experienced odd numbers of Fresnel reflections, which is detected by photodetector PD1.

Fig. 7 Schematic for measurement of the output signals of the hybrid air-core PBF ring resonator.

Download Full Size | PDF

Figure 8 shows the measured results of both the forward and backward signals output from the hybrid air-core PBF ring resonator. The biggest finesse of the forward output signal is about 6.67, and meanwhile the resonant depth is 0.3246, keeping consistent with the simulation results. While the amplitude of the backward output signal gets to the peak, the depths of the forward output signal are relatively small. Considered that the backward output signal is magnified for 30 times at the output port of PD1 for clear observation, and the circulator brings about a loss of 1.35 dB, the biggest depth of the forward output signal is 37.5 times larger than the corresponding peak of the backward output signal. The test results agree well with the simulation.

Fig. 8 Measured results both of the forward and backward signals output from the hybrid air-core PBF ring resonator.

Download Full Size | PDF

The experimental setup of the RFOG equipped with this hybrid PM air-core PBF ring resonator is shown in Fig. 9. All the fibers in the system are PM operation. A lightwave from a 3-kHz linewidth fiber laser is divided into two equivalent beams by coupler C1. The double phase modulation technique was applied [20] to reduce the backward signals from the odd numbers of Fresnel reflections at the two junctions. The LiNbO₃ phase modulators PM1, PM2, PM3 and PM4 are driven by the sinusoidal waves with modulation frequencies f₁, f₂, f₃ and f₄, respectively. Phase modulator PM3 and PM4 are added for the additional carrier suppression. Two circulators CIR1 and CIR2 are used to couple light input to and output from the resonator. The CW and the CCW lightwaves from the resonator are detected by the InGaAs PIN photodetectors, PD1 and PD2. The output of PD1 is fed back through the lock-in amplifier LIA1 to the servo controller PI1 to reduce the reciprocal noises in the RFOG.

Fig. 9 Schematic diagram of the RFOG equipped with the hybrid PM air-core PBF ring resonator.

Download Full Size | PDF

Figure 10(a) shows the open-loop output of the RFOG equipped with this hybrid PM air-core PBF ring resonator for a running time for 3600 seconds with an integration time of 1 s. The peak-to-peak value in one hour is about 0.344°/s, which is close to the short-term precision in 100 s. Thus, the main factor deteriorating the stability of the gyro is the short-term optical noises, mainly affected by the even numbers of Fresnel reflections. Due to strong Fresnel reflections exist in this hybrid ring resonator, the effect from the odd numbers of the reflections has effectively reduced by the double phase modulation technique [20]. However, the influences of the even numbers of reflections still exist. Figure 10(b) further illustrates the nature of the gyro stability, in which the uncertainty in the rotation rate is plotted versus the integration time under a typical data run of 3600 s. The gyro bias stability is about 0.007°/s for an integration time of 400 s are successfully demonstrated, which is the best result, reported to date and to the best of our knowledge, for the air-core PBF ring-resonator gyro. The bias stability can be improved by further reducing the influences of the even numbers of Fresnel reflections at the junctions.

Fig. 10 Measurement results of the open-loop RFOG equipped with the hybrid PM air-core PBF ring resonator for an integration time of 1s. (a) Typical outputs of the stationary RFOG in the running time for an hour. (b) Allan deviation of the open-loop output.

Download Full Size | PDF

4. Conclusion

The air-core PBF has the advantages of low nonlinearity and high temperature stability, which is an ideal choice for the fiber ring resonator applications. A hybrid air-core PBF ring resonator made with a PM optical coupler has been proposed in this article. The effects of the Fresnel reflections caused by junctions between air-core PBFs and solid-core fibers have been theoretically analyzed. The transfer functions of both the forward and backward output signals of the hybrid air-core PBF ring resonator are deduced and verified by experiments. With the application of the double phase modulation technique, the noise from the odd numbers of Fresnel reflections is well suppressed and a bias stability of 0.007°/s is successfully demonstrated in this hybrid PM air-core PBF ring-resonator gyro. Due to the even numbers of Fresnel reflections at the splicing points superimposed on the normal resonant output signal, the bias stability of the hybrid PM air-core PBF ring –resonator gyro is worse than the state-of-the-art RFOG equipped with the polarizing-fiber ring resonator [21]. Although the accuracy of the hybrid PM air-core PBF ring-resonator gyro achieved is lower than the expectation, the potential of air-core PBFs applied in the RFOG is very promising. To improve the performance of the air-core PBF ring-resonator gyro, further investigations are needed to reduce the splice loss and reflections at the junction between the air-core PBF and solid-core fiber.

Acknowledgment

The authors would like to acknowledge financial support from the National Natural Science Foundation of China (NSFC) (No. 61377101).

References and links

1. R. E. Meyer, S. Ezekiel, D. W. Stowe, and V. J. Tekippe, “Passive fiber-optic ring resonator for rotation sensing,” Opt. Lett. 8(12), 644–646 (1983). [CrossRef] [PubMed]

2. F. Zarinetchi, R. E. Meyer, G. A. Sanders, and S. Ezekiel, “Passive resonator gyro,” Proc. SPIE 478, 122–127 (1984). [CrossRef]

3. G. A. Sanders, “Critical review of resonator fiber optic gyroscope technology,” Proc. SPIE 44, 133–159 (1992).

4. K. Iwatsuki, K. Hotate, and M. Higashiguchi, “Effect of Rayleigh backscattering in an optical passive ring-resonator gyro,” Appl. Opt. 23(21), 3916–3924 (1984). [CrossRef] [PubMed]

5. K. Hotate and Y. Kikuchi, “Analysis of the thermo-optically induced bias drift in resonator fiber optic gyro,” Proc. SPIE 4204, 81–88 (2001). [CrossRef]

6. X. Yu, H. Ma, and Z. Jin, “Improving thermal stability of a resonator fiber optic gyro employing a polarizing resonator,” Opt. Express 21(1), 358–369 (2013). [CrossRef] [PubMed]

7. L. K. Strandjord and G. A. Sanders, “Passive stabilization of temperature dependent polarization errors of a polarization-rotating resonator fiber optic gyroscope,” Proc. SPIE 2510, 81–91 (1995). [CrossRef]

8. K. Takiguchi and K. Hotate, “Method to reduce the optical Kerr-effect induced bias in an optical passive ring-resonator gyro,” IEEE Photonics Technol. Lett. 4(2), 203–206 (1992). [CrossRef]

9. H. Ma, X. Li, G. Zhang, and Z. Jin, “Reduction of optical Kerr-effect induced error in a resonant micro-optic gyro by light-intensity feedback technique,” Appl. Opt. 53(16), 3465–3472 (2014). [CrossRef] [PubMed]

10. N. K. T. Photonics, “PM-1550-01 polarisation maintaining PCF,” [Datasheet]. (2013) [Online]. http://www.nktphotonics.com/files/files/PM-1550-01.pdf.

11. V. Dangui, M. J. F. Digonnet, and G. S. Kino, “Laser-driven photonic-bandgap fiber optic gyroscope with negligible Kerr-induced drift,” Opt. Lett. 34(7), 875–877 (2009). [CrossRef] [PubMed]

12. X. Zhao, J. Louveau, J. Chamoun, and M. J. Digonnet, “Thermal Sensitivity of the Birefringence of Air-Core Fibers and Implications for the RFOG,” J. Lightwave Technol. 32(14), 2577–2581 (2014). [CrossRef]

13. G. A. Sanders, L. K. Strandjord, and T. Qiu, “Hollow core fiber optic ring resonator for rotation sensing,” in 18th International Optical Fiber Sensors Conference Technical Digest (Optical Society of America, 2006), paper ME6. [CrossRef]

14. M. A. Terrel, J. F. Digonnet, and S. Fan, “Resonant fiber optic gyroscope using an air-core fiber,” J. Lightwave Technol. 30(7), 931–937 (2012). [CrossRef]

15. L. Feng, X. Ren, X. Deng, and H. Liu, “Analysis of a hollow core photonic bandgap fiber ring resonator based on micro-optical structure,” Opt. Express 20(16), 18202–18208 (2012). [CrossRef] [PubMed]

16. H. Ma, Z. Chen, and Z. Jin, “Single-polarization coupler based on air-core photonic bandgap fibers and implications for resonant fiber optic gyro,” J. Lightwave Technol. 32(1), 46–54 (2014). [CrossRef]

17. H. Ma, X. Yu, and Z. Jin, “Reduction of polarization-fluctuation induced drift in resonator fiber optic gyro by a resonator integrating in-line polarizers,” Opt. Lett. 37(16), 3342–3344 (2012). [CrossRef] [PubMed]

18. K. Z. Aghaie, M. J. Digonnet, and S. Fan, “Optimization of the splice loss between photonic-bandgap fibers and conventional single-mode fibers,” Opt. Lett. 35(12), 1938–1940 (2010). [CrossRef] [PubMed]

19. K. Z. Aghaie and M. J. Digonnet, “Sensitivity limit of a coupled-resonator optical waveguide gyroscope with separate input/output coupling,” J. Opt. Soc. Am. B 32(2), 339–344 (2015). [CrossRef]

20. Z. Jin, G. Zhang, H. Mao, and H. Ma, “Resonator micro optic gyro with double phase modulation technique using an FPGA-based digital processor,” Opt. Commun. 285(5), 645–649 (2012). [CrossRef]

21. T. Qiu, J. Wu, L. K. Strandjord, and G. A. Sanders, “Performance of resonator fiber optic gyroscope using external-cavity laser stabilization and optical filtering,” Proc. SPIE 9157, 91570B (2014).

Effect of Fresnel reflections in a hybrid air-core photonic-bandgap fiber ring-resonator gyro

Abstract

1. Introduction

2. Principle and analysis

3. Experiment and result

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (10)

Equations (8)

Optics Express