- Email: [email protected]
Sensitivity analysis of a fiber ring resonator based on an air-core photonic-bandgap fiber
Optical Fiber Technology 16 (2010) 217–221
Contents lists available at ScienceDirect
Optical Fiber Technology www.elsevier.com/locate/yofte
Sensitivity analysis of a fiber ring resonator based on an air-core photonic-bandgap fiber Diqing Ying *, M.S. Demokan, Xinlu Zhang, Wei Jin Department of Electrical Engineering, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history: Received 31 October 2009 Available online 23 May 2010 Keywords: R-FOG Air-core photonic-bandgap fiber Sensitivity
a b s t r a c t The fiber ring resonator (FRR) is the core sensing element in a resonator fiber optic gyroscope (R-FOG), and its sensitivity determines the performance of the R-FOG. This paper presents an in-depth analysis of the sensitivity of the FRR which is made of an air-core photonic-bandgap fiber (PBF), in which the characteristics of the FRR using PBF are compared with that of an FRR using a conventional single mode fiber. When using PBF instead of conventional fiber, it is found that the resonance curve is changed, and the sensitivity of the FRR is decreased a little when a narrow spectral linewidth laser is used. However, the degree of the decrease in sensitivity is not big enough to deny the advantages of PBF in improving the performance of the R-FOG considering that PBF is much better than conventional fiber in reducing the drift. Also, the optimal parameters of the directional coupler for sensitivity are discussed. It is found that the optimal intensity coupling coefficient when using PBF is nearly two times larger than that when using conventional fiber, and the optimal coupler intensity loss when using PBF is smaller than that when using conventional fiber. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Using the Sagnac effect [1], a resonator fiber optic gyro (R-FOG) has potential as a high accuracy inertial rotation sensor [2]. However, the Rayleigh backscattering [3,4], the Kerr [5,6], Faraday [7,8] and thermal effects [9,10] generally limit the accuracy of the R-FOG. Air-core photonic-bandgap fiber (PBF) is a new kind of fiber [11,12]. Since the optical mode is mostly confined to the air-core when the light travels in the air-core PBF, the four effects in an air-core PBF would be smaller than that in a conventional fiber [11,13]. Therefore, in order to improve the performance of the interference fiber optic gyro (I-FOG), Kim et al. have proposed an I-FOG based on an air-core PBF [13]. The air-core PBF could be used in the R-FOG to improve its performance also; therefore, it is meaningful to study the R-FOG based on an air-core PBF [14,15]. The fiber ring resonator (FRR) is the core sensing element in the R-FOG [16]. The sensitivity of the FRR plays a large role in determining the performance of the R-FOG [17,18]. The analysis of the sensitivity has been done for a FRR using a conventional single mode fiber [17]; however, an in-depth analysis of an FRR based on an air-core PBF has not been done. The characteristic parameters (such as the refractive index and the attenuation loss of the fiber) of the air-core PBF and the conventional fiber are very different, and this would lead to the resonance characteristics of the FRR with different fibers to be different, which would finally * Corresponding author. Fax: +852 23301544. E-mail address: [email protected] (D. Ying). 1068-5200/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.yofte.2010.04.001
give rise to the difference in the sensitivities [1,17]. This paper discusses the sensitivity of a FRR based on an air-core PBF, and a comparison of the sensitivities which occur upon the use of an air-core PBF and conventional fiber are presented. Finally, the optimal parameters of the directional coupler to achieve optimal sensitivity are compared when using PBF and conventional fiber. 2. Theory Fig. 1 illustrates the configuration of the FRR based on an aircore PBF of length L [16]. The input electric field Ein is divided into two parts: one part named EFRR is coupled into the FRR through the directional coupler, and is output from the FRR after circulating in the FRR; and the other part named Ethrough is transmitted through the directional coupler directly. Finally, the interference between Ethrough and EFRR happens at the output port of the directional coupler. In this paper, when we analyze the FRR using PBF, the directional coupler is assumed to be made of air-core PBF also; and when we analyze the FRR using conventional fiber, the directional coupler is assumed to be made of conventional single mode fiber also. Therefore, the large fusion-splice loss between PBF and conventional fiber does not have to be considered. The input electric field Ein can be written as [5,17]:
Ein ðtÞ ¼ E0 expfi½2pf0 t þ uðtÞg
ð1Þ
where E0 is the amplitude of the electric field of the laser light, and f0 is the center frequency of the laser. The phase fluctuation u(t)
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D. Ying et al. / Optical Fiber Technology 16 (2010) 217–221
Directional coupler Light transmitted through the directional coupler directly Ein
Ethrough EFRR Circulated light output from the FRR Light circulating in the FRR Air-core PBF
Fig. 1. Configuration of the FRR based on an air-core PBF.
represents the important parameter of optical source coherence [5], and it satisfies [17,19]:
hexpðiuðtÞÞ expðiuðt sÞÞi ¼ expðpdf sÞ;
s>0
ð2Þ
where df is the spectral linewidth of the laser, and s is the time delay. The output intensity of the FRR normalized by the input intensity can be derived as [17,20]:
" T FRR ¼ ð1 aC Þ 1 q
ð1 Q Þ2
#
2
2
ð1 Q Þ þ 4Q sin ðp Df s0 Þ
ð3Þ
where aC is the intensity loss of the directional coupler; Df = f0 fR is the resonance frequency deviation, fR is the resonance frequency of the FRR; s0 = nrL/c is the transit time in the FRR, nr is the refractive index of the fiber and c is the light velocity in vacuum; and other parameters can be written as [20,21]:
q¼1 T¼
" # 1 2TR ðR0 Þ2 1þQ þ T2 1 aC 1 Q 1 ðQ 0 Þ2 1 Q
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 kC 1 aC
R0 ¼ kC ð1 aC Þ expðaL L=2Þ; Q 0 ¼ expðaL L=2Þ
ð4aÞ
ð4bÞ R ¼ R0 expðpdf s0 Þ
C¼
n
2TRQ ½1ðQ 0 Þ2 2ðR0 Þ2 Q 2 ðTRQ 2 þTRÞ½1ðQ 0 Þ2 ðR0 Þ2 Q ð1þQ 2 Þ
ð4dÞ
o
s0 p
pffiffiffi kP 2C 4A SNR
ð8Þ
where k is the wavelength of the laser, P is the perimeter of the FRR, A is the area enclosed by the FRR, and SNR is the signal-to-noise ratio of the system. The shot noise limited SNR can be written as [17]:
sffiffiffiffiffiffiffiffiffiffiffi gt0 I0 T FRR max T FRR pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SNR ¼ 2hf0 T FRR max
min
ð9Þ
where g is the photo-detector quantum efficiency, t0 is the integration time, I0 ¼ E20 is the intensity of the laser at input of the coupler, and h is Planck’s constant. Since the sensitivity is a function of the resonance characteristic parameters, and the resonance characteristics are dependent on the type of fiber employed, the sensitivities corresponding to PBF and conventional fiber would be different [1,17]. Also, the sensitivity depends on the parameters of the directional coupler kC and aC, therefore, the relationship between the sensitivity, the fiber and the directional coupler parameters will now be looked at closely in order to optimize the characteristics of the FRR. 3. Simulation and discussion
where kC is the intensity coupling coefficient of the directional coupler, and aL is the fiber attenuation loss in the FRR. From Eq. (3), the characteristic parameters of the resonance curve can be obtained. The linewidth of the resonance curve can be derived as [17]:
cos1
dX
ð4cÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 kC 1 aC ;
Q ¼ Q 0 expðpdf s0 Þ
The sensitivity of the FRR is an important parameter of the RFOG, which can be written as [1,10,17]:
ð5Þ
And the maximum and minimum of the resonance curve can be derived as [17]:
T FRR
max
¼ T2 þ
2TR ðR0 Þ2 1Q þ 1 þ Q 1 ðQ 0 Þ2 1 þ Q
ð6Þ
T FRR
min
¼ T2
2TR ðR0 Þ2 1þQ þ 1 Q 1 ðQ 0 Þ2 1 Q
ð7Þ
According to Eqs. (5)–(7), the linewidth, maximum and minimum of the resonance curve are dependent on the characteristic parameters of the fiber. Since the refractive index nr and fiber attenuation loss aL of the PBF and conventional fiber are different, the resonance characteristics of the FRR when using air-core PBF and conventional fiber would also be different.
Fig. 2 illustrates the resonance curve when using air-core PBF or conventional fiber. The parameters are assumed to be as follows: the intensity I0 at the input of the FRR is 1mW, the wavelength of the laser k is 1550 nm, the spectral linewidth of the laser df is 60 kHz, the diameter D of the FRR is 0.1 m, the fiber length L of the FRR is 20 m, the effective refractive index nr is 0.99 for PBF and 1.45 for conventional fiber, the intensity coupling coefficient kC is 5%, the coupler intensity loss aC is 6.67%, and the fiber attenuation loss aL is 20 dB/km for PBF and 0.2 dB/km for conventional fiber [2,14,22]. Comparing the resonance curve when using aircore PBF with that when using conventional fiber, it is obvious that some characteristics of the two resonance curves are different: firstly, the minimum of the resonance curve using air-core PBF is slightly larger than that using conventional fiber; secondly, the linewidth of the resonance curve using air-core PBF is slightly larger than that using conventional fiber. Fig. 3 illustrates the relationship between the sensitivity dX and the spectral linewidth of the laser df when using air-core PBF or conventional fiber. The parameters are as follows: the photo-detector quantum efficiency g is 0.8, the integration time t0 is 1s [5], and other parameters are the same as aforementioned. It is shown in Fig. 3 that the FRR becomes less sensitive as the spectral linewidth of the laser increases. Especially, it is found that the sensitivity using air-core PBF is better than that using conventional fiber only if the linewidth is above a certain value, which is about 600 kHz in
219
0.55
Using air-core PBF Using conventional fiber
1.0
0.50
0.9
Sensitivity δΩ (degree/hour)
Output intensity normalized by input intensity TFRR
D. Ying et al. / Optical Fiber Technology 16 (2010) 217–221
0.8 0.7 0.6 0.5 0.4
Using air-core PBF Using conventional fiber
0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05
0.3
0.00 0.2 6 6 6 5 -2.0x10 -1.5x10 -1.0x10 -5.0x10
0 0.0
5
6
6
2
4
6
Frequency difference between laser frequency and resonance frequency [Δf (Hz)]
8
10 12 14 16 18 20 22 24 26 28 30
Fiber length of the FRR L (m)
6
5.0x10 1.0x10 1.5x10 2.0x10
Fig. 4. The relationship between the sensitivity dX and the fiber length of the FRR L when using air-core PBF or conventional fiber.
Fig. 2. The resonance curve when using air-core PBF or conventional fiber.
magnitude. Because the air-core PBF is much better than conventional fiber in reducing the drifts caused by several important optical effects such as the Kerr effect [14], although the sensitivity is decreased a little when using air-core PBF instead of conventional fiber, the FRR using air-core PBF is still more advantageous than that using conventional fiber in improving the performance of the R-FOG when everything is considered. In the practical R-FOG system, in order to get an optimal sensitivity, the characteristic parameters of the directional coupler are usually optimized [17]. Fig. 5 illustrates the relationship between the sensitivity dX and the intensity coupling coefficient of the directional coupler kC when using air-core PBF or conventional fiber. The other parameters are the same as mentioned in the first and second paragraphs of Section 3. Similar to the FRR using conventional fiber, the FRR using air-core PBF also has an optimal intensity coupling coefficient for the sensitivity, and the optimal intensity coupling coefficient using air-core PBF is different from that using conventional fiber as shown in Fig. 5 [17]. According
0.70 0.65
Using air-core PBF Using conventional fiber
Sensitivity δΩ (degree/hour)
0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0
100
200
300
400
500
600
700
800
900 1000
Spectral linewidth of the laser δf (kHz) 0.7 Fig. 3. The relationship between the sensitivity dX and the spectral linewidth of the laser df when using air-core PBF or conventional fiber.
Fig. 3. However, in order to obtain an optimal sensitivity for the RFOG system, the linewidth is usually very narrow, typically less than 100 kHz [2]. Therefore, in the practical R-FOG system, the sensitivity may be slightly worse when using air-core PBF instead of conventional fiber. Fig. 4 illustrates the relationship between the sensitivity dX and the fiber length of the FRR L when using air-core PBF or conventional fiber. The spectral linewidth of the laser df is assumed to be 60 kHz [2], and other parameters are the same as aforementioned. It is shown in Fig. 4 that the FRR becomes more sensitive as the fiber length increases [5]. Especially, it is found that the sensitivity using air-core PBF is worse than that using conventional fiber because of the narrow spectral linewidth of the laser, which agrees with the result in Fig. 3. However, the sensitivity using air-core PBF is very close to that using conventional fiber. For example, when the fiber length of the FRR is 20 m, the sensitivities are about 0.07 degree/h and 0.03 degree/h for air-core PBF and conventional fiber respectively, and they are of the same order of
Sensitivity δΩ (degree/h)
0.6
Using air-core PBF Using conventional fiber
0.5 0.4 0.3 0.2 0.1 0.0 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Intensity coupling coefficient kC Fig. 5. The relationship between the sensitivity dX and the intensity coupling coefficient of the directional coupler kC when using air-core PBF or conventional fiber.
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D. Ying et al. / Optical Fiber Technology 16 (2010) 217–221
0.20
ical results are helpful to further study and optimize the R-FOG based on the air-core PBF.
Sensitivity δΩ (degree/h)
0.18 0.16
Using air-core PBF Using conventional fiber
Acknowledgment The research was partly supported by the Hong Kong SAR Government through a CERG Grant (PolyU5187/06E).
0.14 0.12
References
0.10 0.08 0.06 0.04 0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Intensity loss of the directional coupler αC Fig. 6. The relationship between the sensitivity dX and the intensity loss of the directional coupler aC when using air-core PBF or conventional fiber.
to the simulation result, the optimal intensity coupling coefficient is about 0.09 for the FRR using air-core PBF and 0.05 for the FRR using conventional fiber. Thus, in order to obtain the optimal sensitivity, the intensity coupling coefficient of the directional coupler when using PBF should be nearly two times larger than that when using conventional fiber. Fig. 6 illustrates the relationship between the sensitivity dX and the intensity loss of the directional coupler aC when using air-core PBF or conventional fiber. The other parameters are the same as mentioned in the first and second paragraphs of Section 3. It is shown in Fig. 6 that the optimal intensity loss is about 0.03 for conventional fiber; however, the sensitivity becomes better and better as the intensity loss approaches zero for PBF. Therefore, in order to obtain the optimal sensitivity when using PBF, the intensity loss of the directional coupler should be as small as possible, and it would generally be smaller than the intensity loss when using conventional fiber.
4. Conclusions The sensitivity characteristics of the FRR based on an air-core PBF has been analyzed, and it is compared with that of the FRR based on a conventional fiber. It is concluded that the resonance curve using PBF is different from that using conventional fiber. The difference of the resonance curves finally causes the difference of the sensitivities, and it is found that the sensitivity would be decreased under narrow spectral linewidth laser when using PBF instead of conventional fiber; however, the degree of the decrease is not big enough to dispel the advantages of the PBF in improving the performance of the R-FOG because the air-core PBF is much better than conventional fiber in reducing several important optical effects such as the Kerr effect. Therefore, using air-core PBF instead of conventional fiber still has the potential to improve the performance of the R-FOG significantly. In addition, the optimal parameters of the directional coupler for sensitivity are considered, and it is found that the optimal intensity coupling coefficient when using PBF is nearly two times larger than that when using conventional fiber, and the optimal coupler intensity loss when using PBF is smaller than that when using conventional fiber. These theoret-
[1] W.W. Chow, J. Gea-Banaciloche, L.M. Pedrotti, The ring laser gyro, Rev. Mod. Phys. 57 (1985) 61–103. [2] Diqing Ying, Huilian Ma, Zhonghe Jin, Resonator fiber optic gyro using the triangle wave phase modulation technique, Opt. Commun. 281 (2008) 580–586. [3] K. Iwatsuki, K. Hotate, M. Higashiguchi, Resonance characteristics of backscattering in optical passive-ring resonator gyro: experiment, Appl. Optics 25 (1986) 4448–4451. [4] K. Iwatsuki, K. Hotate, M. Higashiguchi, Effect of Rayleigh backscattering in an optical passive ring-resonator Gyro, Appl. Optics 23 (1984) 3916–3924. [5] K. Iwatsuki, K. Hotate, M. Higashiguchi, Kerr effect in an optical passive ringresonator gyro, J. Lightw. Technol. LT-4 (1986) 645–651. [6] K. Takiguchi, K. Hotate, Method to reduce the optical Kerr-effect-induced bias in an optical passive ring-resonator gyro, IEEE Photon. Technol. Lett. 4 (1992) 203–206. [7] K. Hotate, K. Tabe, Drift of an optical fiber gyroscope caused by the Faraday effect: experiment, J. Lightw. Technol. LT-5 (1987) 997–1001. [8] T. Ito, K. Hotate, Closed-loop operation in the resonator fiber optic gyro using Faraday effect with a twisted single-mode-fiber resonator, in: SPIE 2837, 1996, pp. 260–271. [9] K. Hotate, Y. Kikuchi, Analysis of thermooptically induced bias drift in resonator fiber optic gyro, in: SPIE 4204, 2001, pp. 81–88. [10] D.M. Shupe, Fiber resonator gyroscope: sensitivity and thermal nonreciprocity, Appl. Optics 20 (1981) 286–289. [11] C.M. Smith, N. Venkataraman, M.T. Gallagher, D. Muller, J.A. West, N.F. Borrelli, D.C. Allan, K.W. Koch, Low-loss hollow-core silica/air photonic bandgap fibre, Nature 424 (2003) 657–659. [12] R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, D.C. Allan, Single-mode photonic band gap guidance of light in air, Science 285 (1999) 1537–1539. [13] H.K. Kim, M.J.F. Digonnet, G.S. Kino, Air-core photonic-bandgap fiber–optic gyroscope, J. Lightw. Technol. 24 (2006) 3169–3174. [14] Y. Xiujuan, L. Yanbiao, Z. Min, Y. Youlong, L. Dejie, Kerr effect in an optical passive ring-resonator gyro using a hollow-core photonic band-gap fiber, Chin. J. Lasers 35 (2008) 430–435. [15] G.A. Sanders, L.K. Strandjord, T. Qiu, Hollow core fiber optic ring resonator for rotation sensing, in: Optical Fiber Sensors, OSA Technical Digest (CD), Optical Society of America, 2006. paper ME6. [16] Diqing Ying, Huilian Ma, Zhonghe Jin, Ringing phenomenon of the fiber ring resonator, Appl. Optics 46 (2007) 4890–4895. [17] Huilian Ma, Jin Zhonghe, Ding Chun, Wang Yuelin, Influence of spectral linewidth of laser on resonance characteristics in fiber ring resonator, Chin. J. Lasers 30 (2003) 731–734. [18] K. Hotate, M. Harumoto, Resonator fiber optic gyro using digital serrodyne modulation, J. Lightw. Technol. 15 (1997) 466–473. [19] Y. Ohtsuka, Optical coherence effects on a fiber-sensing Fabry-Perot interferometer, Appl. Optics 21 (1982) 4316–4320. [20] Huilian Ma, Jin Zhonghe, Ding Chun, Wang Yuelin, Research on signal detection method of resonator fiber optical gyro, Chin. J. Lasers 31 (2004) 1001–1005. [21] Zhang Xulin, Huilian Ma, Ding Chun, Jin Zhonghe, Optical Kerr effect in phase modulation spectroscopy resonator fiber optic gyro, Chin. J. Lasers 33 (2006) 814–818. [22] Diqing Ying, Huilian Ma, Zhonghe Jin, Dynamic characteristics of R-FOG based on the triangle wave phase modulation technique, Opt. Commun. 281 (2008) 5340–5343.
Diqing Ying, born in Zhejiang Province, P.R. China, on June 5, 1980, received his B.E. and Ph.D. degree from Zhejiang University in 2003 and 2008, respectively. He is currently a postdoctoral fellow in Hong Kong Polytechnic University. His research interests are in optical fiber sensors. Email: [email protected],hk, tel.: +852 51287206, fax: +852 23301544.
D. Ying et al. / Optical Fiber Technology 16 (2010) 217–221 M.S. Demokan received the B.Sc. degree in electronic engineering from the Middle East Technical University, Ankara, Turkey, in 1970, and the M.Sc. and Ph.D. degrees in electronic engineering from King’s College, University of London, London, UK, in 1972 and 1976, respectively. He is currently the Vice-President responsible for academic development in Hong Kong Polytechnic \University. His research interests include optical communication systems (especially all-optical switching and photonic crystal fibers) and various types of optical sensors.
Xinlu Zhang was born in Heilongjiang, China, in 1971. He received the B.S. degree in physics from Jinlin University in 1995, the M.S. and Ph.D. degrees in physics electronics from Harbin institute of technology in 2001 and 2005 respectively. Since 2005, he has been a professor at the College of Science, Harbin Engineering University, Heilongjiang, China. His research interests are diode-pumped solid state laser and nonlinear optics. He has published more than 60 scientific papers.
221 Wei Jin received his B.Eng and M.Sc. degrees from Beijing University of Aeronautics and Astronautics in 1984 and 1987, respectively. He received the Ph.D. degree in 1991 in fiber optics from University of Strathclyde and afterwards was employed as a Postdoctoral Research Fellow at the same University till the end of 1995. He is currently the professor of electrical engineering department in Hong Kong Polytechnic University. His research interests are photonic crystal fibers and devices, optical fiber sensors, fiber lasers and amplifiers, optical gas detectors, condition monitoring of electrical power transformers, and civil and mechanical structures.
Contents lists available at ScienceDirect
Optical Fiber Technology www.elsevier.com/locate/yofte
Sensitivity analysis of a fiber ring resonator based on an air-core photonic-bandgap fiber Diqing Ying *, M.S. Demokan, Xinlu Zhang, Wei Jin Department of Electrical Engineering, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong
a r t i c l e
i n f o
Article history: Received 31 October 2009 Available online 23 May 2010 Keywords: R-FOG Air-core photonic-bandgap fiber Sensitivity
a b s t r a c t The fiber ring resonator (FRR) is the core sensing element in a resonator fiber optic gyroscope (R-FOG), and its sensitivity determines the performance of the R-FOG. This paper presents an in-depth analysis of the sensitivity of the FRR which is made of an air-core photonic-bandgap fiber (PBF), in which the characteristics of the FRR using PBF are compared with that of an FRR using a conventional single mode fiber. When using PBF instead of conventional fiber, it is found that the resonance curve is changed, and the sensitivity of the FRR is decreased a little when a narrow spectral linewidth laser is used. However, the degree of the decrease in sensitivity is not big enough to deny the advantages of PBF in improving the performance of the R-FOG considering that PBF is much better than conventional fiber in reducing the drift. Also, the optimal parameters of the directional coupler for sensitivity are discussed. It is found that the optimal intensity coupling coefficient when using PBF is nearly two times larger than that when using conventional fiber, and the optimal coupler intensity loss when using PBF is smaller than that when using conventional fiber. Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction Using the Sagnac effect [1], a resonator fiber optic gyro (R-FOG) has potential as a high accuracy inertial rotation sensor [2]. However, the Rayleigh backscattering [3,4], the Kerr [5,6], Faraday [7,8] and thermal effects [9,10] generally limit the accuracy of the R-FOG. Air-core photonic-bandgap fiber (PBF) is a new kind of fiber [11,12]. Since the optical mode is mostly confined to the air-core when the light travels in the air-core PBF, the four effects in an air-core PBF would be smaller than that in a conventional fiber [11,13]. Therefore, in order to improve the performance of the interference fiber optic gyro (I-FOG), Kim et al. have proposed an I-FOG based on an air-core PBF [13]. The air-core PBF could be used in the R-FOG to improve its performance also; therefore, it is meaningful to study the R-FOG based on an air-core PBF [14,15]. The fiber ring resonator (FRR) is the core sensing element in the R-FOG [16]. The sensitivity of the FRR plays a large role in determining the performance of the R-FOG [17,18]. The analysis of the sensitivity has been done for a FRR using a conventional single mode fiber [17]; however, an in-depth analysis of an FRR based on an air-core PBF has not been done. The characteristic parameters (such as the refractive index and the attenuation loss of the fiber) of the air-core PBF and the conventional fiber are very different, and this would lead to the resonance characteristics of the FRR with different fibers to be different, which would finally * Corresponding author. Fax: +852 23301544. E-mail address: [email protected] (D. Ying). 1068-5200/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.yofte.2010.04.001
give rise to the difference in the sensitivities [1,17]. This paper discusses the sensitivity of a FRR based on an air-core PBF, and a comparison of the sensitivities which occur upon the use of an air-core PBF and conventional fiber are presented. Finally, the optimal parameters of the directional coupler to achieve optimal sensitivity are compared when using PBF and conventional fiber. 2. Theory Fig. 1 illustrates the configuration of the FRR based on an aircore PBF of length L [16]. The input electric field Ein is divided into two parts: one part named EFRR is coupled into the FRR through the directional coupler, and is output from the FRR after circulating in the FRR; and the other part named Ethrough is transmitted through the directional coupler directly. Finally, the interference between Ethrough and EFRR happens at the output port of the directional coupler. In this paper, when we analyze the FRR using PBF, the directional coupler is assumed to be made of air-core PBF also; and when we analyze the FRR using conventional fiber, the directional coupler is assumed to be made of conventional single mode fiber also. Therefore, the large fusion-splice loss between PBF and conventional fiber does not have to be considered. The input electric field Ein can be written as [5,17]:
Ein ðtÞ ¼ E0 expfi½2pf0 t þ uðtÞg
ð1Þ
where E0 is the amplitude of the electric field of the laser light, and f0 is the center frequency of the laser. The phase fluctuation u(t)
218
D. Ying et al. / Optical Fiber Technology 16 (2010) 217–221
Directional coupler Light transmitted through the directional coupler directly Ein
Ethrough EFRR Circulated light output from the FRR Light circulating in the FRR Air-core PBF
Fig. 1. Configuration of the FRR based on an air-core PBF.
represents the important parameter of optical source coherence [5], and it satisfies [17,19]:
hexpðiuðtÞÞ expðiuðt sÞÞi ¼ expðpdf sÞ;
s>0
ð2Þ
where df is the spectral linewidth of the laser, and s is the time delay. The output intensity of the FRR normalized by the input intensity can be derived as [17,20]:
" T FRR ¼ ð1 aC Þ 1 q
ð1 Q Þ2
#
2
2
ð1 Q Þ þ 4Q sin ðp Df s0 Þ
ð3Þ
where aC is the intensity loss of the directional coupler; Df = f0 fR is the resonance frequency deviation, fR is the resonance frequency of the FRR; s0 = nrL/c is the transit time in the FRR, nr is the refractive index of the fiber and c is the light velocity in vacuum; and other parameters can be written as [20,21]:
q¼1 T¼
" # 1 2TR ðR0 Þ2 1þQ þ T2 1 aC 1 Q 1 ðQ 0 Þ2 1 Q
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 kC 1 aC
R0 ¼ kC ð1 aC Þ expðaL L=2Þ; Q 0 ¼ expðaL L=2Þ
ð4aÞ
ð4bÞ R ¼ R0 expðpdf s0 Þ
C¼
n
2TRQ ½1ðQ 0 Þ2 2ðR0 Þ2 Q 2 ðTRQ 2 þTRÞ½1ðQ 0 Þ2 ðR0 Þ2 Q ð1þQ 2 Þ
ð4dÞ
o
s0 p
pffiffiffi kP 2C 4A SNR
ð8Þ
where k is the wavelength of the laser, P is the perimeter of the FRR, A is the area enclosed by the FRR, and SNR is the signal-to-noise ratio of the system. The shot noise limited SNR can be written as [17]:
sffiffiffiffiffiffiffiffiffiffiffi gt0 I0 T FRR max T FRR pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SNR ¼ 2hf0 T FRR max
min
ð9Þ
where g is the photo-detector quantum efficiency, t0 is the integration time, I0 ¼ E20 is the intensity of the laser at input of the coupler, and h is Planck’s constant. Since the sensitivity is a function of the resonance characteristic parameters, and the resonance characteristics are dependent on the type of fiber employed, the sensitivities corresponding to PBF and conventional fiber would be different [1,17]. Also, the sensitivity depends on the parameters of the directional coupler kC and aC, therefore, the relationship between the sensitivity, the fiber and the directional coupler parameters will now be looked at closely in order to optimize the characteristics of the FRR. 3. Simulation and discussion
where kC is the intensity coupling coefficient of the directional coupler, and aL is the fiber attenuation loss in the FRR. From Eq. (3), the characteristic parameters of the resonance curve can be obtained. The linewidth of the resonance curve can be derived as [17]:
cos1
dX
ð4cÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 kC 1 aC ;
Q ¼ Q 0 expðpdf s0 Þ
The sensitivity of the FRR is an important parameter of the RFOG, which can be written as [1,10,17]:
ð5Þ
And the maximum and minimum of the resonance curve can be derived as [17]:
T FRR
max
¼ T2 þ
2TR ðR0 Þ2 1Q þ 1 þ Q 1 ðQ 0 Þ2 1 þ Q
ð6Þ
T FRR
min
¼ T2
2TR ðR0 Þ2 1þQ þ 1 Q 1 ðQ 0 Þ2 1 Q
ð7Þ
According to Eqs. (5)–(7), the linewidth, maximum and minimum of the resonance curve are dependent on the characteristic parameters of the fiber. Since the refractive index nr and fiber attenuation loss aL of the PBF and conventional fiber are different, the resonance characteristics of the FRR when using air-core PBF and conventional fiber would also be different.
Fig. 2 illustrates the resonance curve when using air-core PBF or conventional fiber. The parameters are assumed to be as follows: the intensity I0 at the input of the FRR is 1mW, the wavelength of the laser k is 1550 nm, the spectral linewidth of the laser df is 60 kHz, the diameter D of the FRR is 0.1 m, the fiber length L of the FRR is 20 m, the effective refractive index nr is 0.99 for PBF and 1.45 for conventional fiber, the intensity coupling coefficient kC is 5%, the coupler intensity loss aC is 6.67%, and the fiber attenuation loss aL is 20 dB/km for PBF and 0.2 dB/km for conventional fiber [2,14,22]. Comparing the resonance curve when using aircore PBF with that when using conventional fiber, it is obvious that some characteristics of the two resonance curves are different: firstly, the minimum of the resonance curve using air-core PBF is slightly larger than that using conventional fiber; secondly, the linewidth of the resonance curve using air-core PBF is slightly larger than that using conventional fiber. Fig. 3 illustrates the relationship between the sensitivity dX and the spectral linewidth of the laser df when using air-core PBF or conventional fiber. The parameters are as follows: the photo-detector quantum efficiency g is 0.8, the integration time t0 is 1s [5], and other parameters are the same as aforementioned. It is shown in Fig. 3 that the FRR becomes less sensitive as the spectral linewidth of the laser increases. Especially, it is found that the sensitivity using air-core PBF is better than that using conventional fiber only if the linewidth is above a certain value, which is about 600 kHz in
219
0.55
Using air-core PBF Using conventional fiber
1.0
0.50
0.9
Sensitivity δΩ (degree/hour)
Output intensity normalized by input intensity TFRR
D. Ying et al. / Optical Fiber Technology 16 (2010) 217–221
0.8 0.7 0.6 0.5 0.4
Using air-core PBF Using conventional fiber
0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05
0.3
0.00 0.2 6 6 6 5 -2.0x10 -1.5x10 -1.0x10 -5.0x10
0 0.0
5
6
6
2
4
6
Frequency difference between laser frequency and resonance frequency [Δf (Hz)]
8
10 12 14 16 18 20 22 24 26 28 30
Fiber length of the FRR L (m)
6
5.0x10 1.0x10 1.5x10 2.0x10
Fig. 4. The relationship between the sensitivity dX and the fiber length of the FRR L when using air-core PBF or conventional fiber.
Fig. 2. The resonance curve when using air-core PBF or conventional fiber.
magnitude. Because the air-core PBF is much better than conventional fiber in reducing the drifts caused by several important optical effects such as the Kerr effect [14], although the sensitivity is decreased a little when using air-core PBF instead of conventional fiber, the FRR using air-core PBF is still more advantageous than that using conventional fiber in improving the performance of the R-FOG when everything is considered. In the practical R-FOG system, in order to get an optimal sensitivity, the characteristic parameters of the directional coupler are usually optimized [17]. Fig. 5 illustrates the relationship between the sensitivity dX and the intensity coupling coefficient of the directional coupler kC when using air-core PBF or conventional fiber. The other parameters are the same as mentioned in the first and second paragraphs of Section 3. Similar to the FRR using conventional fiber, the FRR using air-core PBF also has an optimal intensity coupling coefficient for the sensitivity, and the optimal intensity coupling coefficient using air-core PBF is different from that using conventional fiber as shown in Fig. 5 [17]. According
0.70 0.65
Using air-core PBF Using conventional fiber
Sensitivity δΩ (degree/hour)
0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0
100
200
300
400
500
600
700
800
900 1000
Spectral linewidth of the laser δf (kHz) 0.7 Fig. 3. The relationship between the sensitivity dX and the spectral linewidth of the laser df when using air-core PBF or conventional fiber.
Fig. 3. However, in order to obtain an optimal sensitivity for the RFOG system, the linewidth is usually very narrow, typically less than 100 kHz [2]. Therefore, in the practical R-FOG system, the sensitivity may be slightly worse when using air-core PBF instead of conventional fiber. Fig. 4 illustrates the relationship between the sensitivity dX and the fiber length of the FRR L when using air-core PBF or conventional fiber. The spectral linewidth of the laser df is assumed to be 60 kHz [2], and other parameters are the same as aforementioned. It is shown in Fig. 4 that the FRR becomes more sensitive as the fiber length increases [5]. Especially, it is found that the sensitivity using air-core PBF is worse than that using conventional fiber because of the narrow spectral linewidth of the laser, which agrees with the result in Fig. 3. However, the sensitivity using air-core PBF is very close to that using conventional fiber. For example, when the fiber length of the FRR is 20 m, the sensitivities are about 0.07 degree/h and 0.03 degree/h for air-core PBF and conventional fiber respectively, and they are of the same order of
Sensitivity δΩ (degree/h)
0.6
Using air-core PBF Using conventional fiber
0.5 0.4 0.3 0.2 0.1 0.0 0.00
0.05
0.10
0.15
0.20
0.25
0.30
Intensity coupling coefficient kC Fig. 5. The relationship between the sensitivity dX and the intensity coupling coefficient of the directional coupler kC when using air-core PBF or conventional fiber.
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D. Ying et al. / Optical Fiber Technology 16 (2010) 217–221
0.20
ical results are helpful to further study and optimize the R-FOG based on the air-core PBF.
Sensitivity δΩ (degree/h)
0.18 0.16
Using air-core PBF Using conventional fiber
Acknowledgment The research was partly supported by the Hong Kong SAR Government through a CERG Grant (PolyU5187/06E).
0.14 0.12
References
0.10 0.08 0.06 0.04 0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
Intensity loss of the directional coupler αC Fig. 6. The relationship between the sensitivity dX and the intensity loss of the directional coupler aC when using air-core PBF or conventional fiber.
to the simulation result, the optimal intensity coupling coefficient is about 0.09 for the FRR using air-core PBF and 0.05 for the FRR using conventional fiber. Thus, in order to obtain the optimal sensitivity, the intensity coupling coefficient of the directional coupler when using PBF should be nearly two times larger than that when using conventional fiber. Fig. 6 illustrates the relationship between the sensitivity dX and the intensity loss of the directional coupler aC when using air-core PBF or conventional fiber. The other parameters are the same as mentioned in the first and second paragraphs of Section 3. It is shown in Fig. 6 that the optimal intensity loss is about 0.03 for conventional fiber; however, the sensitivity becomes better and better as the intensity loss approaches zero for PBF. Therefore, in order to obtain the optimal sensitivity when using PBF, the intensity loss of the directional coupler should be as small as possible, and it would generally be smaller than the intensity loss when using conventional fiber.
4. Conclusions The sensitivity characteristics of the FRR based on an air-core PBF has been analyzed, and it is compared with that of the FRR based on a conventional fiber. It is concluded that the resonance curve using PBF is different from that using conventional fiber. The difference of the resonance curves finally causes the difference of the sensitivities, and it is found that the sensitivity would be decreased under narrow spectral linewidth laser when using PBF instead of conventional fiber; however, the degree of the decrease is not big enough to dispel the advantages of the PBF in improving the performance of the R-FOG because the air-core PBF is much better than conventional fiber in reducing several important optical effects such as the Kerr effect. Therefore, using air-core PBF instead of conventional fiber still has the potential to improve the performance of the R-FOG significantly. In addition, the optimal parameters of the directional coupler for sensitivity are considered, and it is found that the optimal intensity coupling coefficient when using PBF is nearly two times larger than that when using conventional fiber, and the optimal coupler intensity loss when using PBF is smaller than that when using conventional fiber. These theoret-
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Diqing Ying, born in Zhejiang Province, P.R. China, on June 5, 1980, received his B.E. and Ph.D. degree from Zhejiang University in 2003 and 2008, respectively. He is currently a postdoctoral fellow in Hong Kong Polytechnic University. His research interests are in optical fiber sensors. Email: [email protected],hk, tel.: +852 51287206, fax: +852 23301544.
D. Ying et al. / Optical Fiber Technology 16 (2010) 217–221 M.S. Demokan received the B.Sc. degree in electronic engineering from the Middle East Technical University, Ankara, Turkey, in 1970, and the M.Sc. and Ph.D. degrees in electronic engineering from King’s College, University of London, London, UK, in 1972 and 1976, respectively. He is currently the Vice-President responsible for academic development in Hong Kong Polytechnic \University. His research interests include optical communication systems (especially all-optical switching and photonic crystal fibers) and various types of optical sensors.
Xinlu Zhang was born in Heilongjiang, China, in 1971. He received the B.S. degree in physics from Jinlin University in 1995, the M.S. and Ph.D. degrees in physics electronics from Harbin institute of technology in 2001 and 2005 respectively. Since 2005, he has been a professor at the College of Science, Harbin Engineering University, Heilongjiang, China. His research interests are diode-pumped solid state laser and nonlinear optics. He has published more than 60 scientific papers.
221 Wei Jin received his B.Eng and M.Sc. degrees from Beijing University of Aeronautics and Astronautics in 1984 and 1987, respectively. He received the Ph.D. degree in 1991 in fiber optics from University of Strathclyde and afterwards was employed as a Postdoctoral Research Fellow at the same University till the end of 1995. He is currently the professor of electrical engineering department in Hong Kong Polytechnic University. His research interests are photonic crystal fibers and devices, optical fiber sensors, fiber lasers and amplifiers, optical gas detectors, condition monitoring of electrical power transformers, and civil and mechanical structures.